binomial experiment successes

Binomial Distribution Calculator

What is the binomial probability, binomial probability formula, how to use the binomial distribution calculator: an example, how to calculate cumulative probabilities, binomial probability distribution experiments, mean and variance of binomial distribution, other considerations.

This binomial distribution calculator is here to help you with probability problems in the following form: what is the probability of a certain number of successes in a sequence of events? Read on to learn what exactly is the binomial probability distribution, when and how to apply it, and learn the binomial probability formula. Find out what is binomial distribution, and discover how binomial experiments are used in various settings.

Imagine you're playing a game of dice. To win, you need exactly three out of five dice to show a result equal to or lower than 4. The remaining two dice need to show a higher number. What is the probability of you winning?

This is a sample problem that can be solved with our binomial probability calculator. You know the number of events (it is equal to the total number of dice, so five); you know the number of successes you need (precisely 3); you also can calculate the probability of one single success occurring (4 out of 6, so 0.667). This is all the data required to find the binomial probability of you winning the game of dice.

Note that to use the binomial distribution calculator effectively, the events you analyze must be independent . It means that all the trials in your example are supposed to be mutually exclusive.

The first trial's success doesn't affect the probability of success or the probability of failure in subsequent events, and they stay precisely the same. In the case of a dice game, these conditions are met: each time you roll a die constitutes an independent event.

Sometimes you may be interested in the number of trials you need to achieve a particular outcome. For instance, you may wonder how many rolls of a die are necessary before you throw a six three times. Such questions may be addressed using a related statistical tool called the negative binomial distribution. Make sure to learn about it with Omni's negative binomial distribution calculator .

Also, you may check our normal approximation to binomial distribution calculator and the related continuity correction calculator.

To find this probability, you need to use the following equation:

P(X=r) = nCr × p r × (1-p) n-r

• n – Total number of events;
• r – Number of required successes;
• p  – Probability of one success;
• nCr – Number of combinations (so-called "n choose r"); and
• P(X=r) – Probability of an exact number of successes happening.

You should note that the result is the probability of an exact number of successes. For example, in our game of dice, we needed precisely three successes – no less, no more. What would happen if we changed the rules so that you need at least three successes? Well, you would have to calculate the probability of exactly three, precisely four, and precisely five successes and sum all of these values together.

Let's solve the problem of the game of dice together.

Determine the number of events. n is equal to 5, as we roll five dice.

Determine the required number of successes. r is equal to 3, as we need exactly three successes to win the game.

The probability of rolling 1, 2, 3, or 4 on a six-sided die is 4 out of 6, or 0.667. Therefore p is equal to 0.667 or 66.7%.

Calculate the number of combinations (5 choose 3). You can use the combination calculator to do it. This number, in our case, is equal to 10.

Substitute all these values into the binomial probability formula above:

P(X = 3) = 10 × 0.667 3 × (1-0.667) (5-3) = 10 × 0.667 3 × (1-0.667) (5-3) = 10 × 0.296 × 0.333 2 = 2.96 × 0.111 = 0.329

You can also save yourself some time and use the binomial distribution calculator instead :)

Sometimes, instead of an exact number of successes, you want to know the probability of getting r or more successes or r or less successes. To calculate the probability of getting any range of successes:

• Use the binomial probability formula to calculate the probability of success (P) for all possible values of r you are interested in.
• Sum the values of P for all r within the range of interest.

For example, the probability of getting two or fewer successes when flipping a coin four times (p = 0.5 and n = 4) would be:

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

P(X ≤ 2) = 37.5% + 25% + 6.25%

P(X ≤ 2) = 68.75%

This calculation is made easy using the options available on the binomial distribution calculator. You can change the settings to calculate the probability of getting:

• Exactly r successes: P(X = r)
• r or more successes: P(X ≥ r)
• r or fewer successes: P(X ≤ r)
• Between r₀ and r₁ successes P(r₀ ≤ X ≤ r₁)

The binomial distribution turns out to be very practical in experimental settings . However, the output of such a random experiment needs to be binary : pass or failure, present or absent, compliance or refusal. It's impossible to use this design when there are three possible outcomes.

At the same time, apart from rolling dice or tossing a coin, it may be employed in somehow less clear cases. Here are a couple of questions you can answer with the binomial probability distribution:

• Will a new drug work on a randomly selected patient?
• Will a light bulb you just bought work properly, or will it be broken?
• What is a chance of correctly answering a test question you just drew?
• What is a probability of a random voter to vote for a candidate in an election?
• How likely is it for a group of students to be accepted to a prestigious college?

Experiments with precisely two possible outcomes, such as the ones above, are typical binomial distribution examples, often called the Bernoulli trials .

In practice, you can often find the binomial probability examples in fields like quality control , where this method is used to test the efficiency of production processes. The inspection process based on the binomial distribution is designed to perform a sufficient number of checkups and minimize the chances of manufacturing a defective product.

If you don't know the probability of an independent event in your experiment ( p ), collect the past data in one of your binomial distribution examples, and divide the number of successes ( y ) by the overall number of events p = y/n .

Once you have determined your rate of success (or failure) in a single event, you need to decide what's your acceptable number of successes (or failures) in the long run. For example, one defective product in a batch of fifty is not a tragedy, but you wouldn't like to have every second product faulty, would you?

Bernoulli trials are also perfect at solving network systems . Interestingly, they may be used to work out paths between two nodes on a diagram. This is the case of the Wheatstone bridge network, a representation of a circuit built for electrical resistance measurement.

Like the binomial distribution table , our calculator produces results that help you assess the chances that you will meet your target. However, if you like, you may take a look at this binomial distribution table . It tells you what is the binomial distribution value for a given probability and number of successes.

One of the most exciting features of binomial distributions is that they represent the sum of a number n of independent events. Each of them ( Z ) may assume the values of 0 or 1 over a given period.

Let's say the probability that each Z occurs is p . Since the events are not correlated, we can use random variables' addition properties to calculate the mean (expected value) of the binomial distribution μ = np .

The variance of a binomial distribution is given as: σ² = np(1-p) . The larger the variance, the greater the fluctuation of a random variable from its mean. A small variance indicates that the results we get are spread out over a narrower range of values.

The standard deviation of binomial distribution, another measure of a probability distribution dispersion, is simply the square root of the variance, σ . Keep in mind that the standard deviation calculated from your sample (the observations you actually gather) may differ from the entire population's standard deviation. If you find this distinction confusing, there here's a great explanation of this distinction .

There's a clear-cut intuition behind these formulas. Suppose this time that I flip a coin 20 times:

• My p is then equal to 0.5 (unless, of course, the coin is rigged);
• Each Z has an equivalent chance of 0 or 1;
• The number of trials, n , is 20.

This sequence of events fulfills the prerequisites of a binomial distribution.

The mean value of this simple experiment is: np = 20 × 0.5 = 10 . We can say that on average if we repeat the experiment many times, we should expect heads to appear ten times.

The variance of this binomial distribution is equal to np(1-p) = 20 × 0.5 × (1-0.5) = 5 . Take the square root of the variance, and you get the standard deviation of the binomial distribution, 2.24 . Accordingly, the typical results of such an experiment will deviate from its mean value by around 2. Hence, in most of the trials, we expect to get anywhere from 8 to 12 successes.

Use our binomial probability calculator to get the mean, variance, and standard deviation of binomial distribution based on the number of events you provided and the probability of one success.

Developed by a Swiss mathematician Jacob Bernoulli , the binomial distribution is a more general formulation of the Poisson distribution. In the latter, we simply assume that the number of events (trials) is enormous, but the probability of a single success is small.

The binomial distribution is closely related to the binomial theorem , which proves to be useful for computing permutations and combinations. Make sure to check out our permutations calculator , too!

Keep in mind that the binomial distribution formula describes a discrete distribution . The possible outcomes of all the trials must be distinct and non-overlapping. What's more, the two outcomes of an event must be complementary: for a given p , there's always an event of q = 1-p .

If there's a chance of getting a result between the two, such as 0.5, the binomial distribution formula should not be used. The same goes for the outcomes that are non-binary, e.g., an effect in your experiment may be classified as low, moderate, or high.

However, for a sufficiently large number of trials, the binomial distribution formula may be approximated by the Gaussian (normal) distribution specification, with a given mean and variance. That allows us to perform the so-called continuity correction , and account for non-integer arguments in the probability function.

Maybe you still need some practice with the binomial probability distribution examples?

Try to solve the dice game's problem again, but this time you need three or more successes to win it. How about the chances of getting exactly 4?

Is the binomial distribution discrete or continuous?

The binomial distribution is discrete – it takes only a finite number of values.

How do I find the mean of a binomial distribution?

To calculate the mean (expected value) of a binomial distribution B(n,p) you need to multiply the number of trials n by the probability of successes p , that is: mean = n × p .

How do I find the standard deviation of a binomial distribution?

To find the standard deviation of a binomial distribution B(n,p) :

• Compute the variance as n × p × (1-p) , where n is the number of trials and p is the probability of successes.
• Take the square root of the number obtained in Step 1.
• That's it! Congrats :)

What is the probability of 3 successes in 5 trials if the probability of success is 0.5?

To find this probability, you need to:

Recall the binomial distribution formula P(X = r) = nCr × pʳ × (1-p)ⁿ⁻ʳ . We'll use it with the following data:

Number of trials: n = 5 ;

Number of successes: r = 3 ; and

Probability of success: p = 0.5 .

Calculate 5 choose 3 : nCr = 10 .

Plug these values into the formula:

P(X = 3) = 10 × 0.5² × 0.5³ = 0.3125 .

The probability you're looking for is 31.25% .

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Binomial Distribution

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The binomial distribution is, in essence, the probability distribution of the number of heads resulting from flipping a weighted coin multiple times. It is useful for analyzing the results of repeated independent trials, especially the probability of meeting a particular threshold given a specific error rate, and thus has applications to risk management . For this reason, the binomial distribution is also important in determining statistical significance .

Formal Definition

Finding the binomial distribution, properties of the binomial distribution, practical applications, binomial test.

A Bernoulli trial , or Bernoulli experiment , is an experiment satisfying two key properties:

• There are exactly two complementary outcomes, success and failure.
• The probability of success is the same every time the experiment is repeated.

A binomial experiment is a series of \(n\) Bernoulli trials, whose outcomes are independent of each other. A random variable , \(X\), is defined as the number of successes in a binomial experiment. Finally, a binomial distribution is the probability distribution of \(X\).

For example, consider a fair coin. Flipping the coin once is a Bernoulli trial, since there are exactly two complementary outcomes (flipping a head and flipping a tail), and they are both \(\frac{1}{2}\) no matter how many times the coin is flipped. Note that the fact that the coin is fair is not necessary; flipping a weighted coin is still a Bernoulli trial.

A binomial experiment might consist of flipping the coin 100 times, with the resulting number of heads being represented by the random variable \(X\). The binomial distribution of this experiment is the probability distribution of \(X.\)

Determining the binomial distribution is straightforward but computationally tedious. If there are \(n\) Bernoulli trials, and each trial has a probability \(p\) of success, then the probability of exactly \(k\) successes is

\[\binom{n}{k}p^k(1-p)^{n-k}.\]

This is written as \(\text{Pr}(X=k)\), denoting the probability that the random variable \(X\) is equal to \(k\), or as \(b(k;n,p)\), denoting the binomial distribution with parameters \(n\) and \(p\).

The above formula is derived from choosing exactly \(k\) of the \(n\) trials to result in successes, for which there are \(\binom{n}{k}\) choices, then accounting for the fact that each of the trials marked for success has a probability \(p\) of resulting in success, and each of the trials marked for failure has a probability \(1-p\) of resulting in failure. The binomial coefficient \(\binom{n}{k}\) lends its name to the binomial distribution.

Consider a weighted coin that flips heads with probability \(0.25\). If the coin is flipped 5 times, what is the resulting binomial distribution? This binomial experiment consists of 5 trials, a \(p\)-value of \(0.25\), and the number of successes is either 0, 1, 2, 3, 4, or 5. Therefore, the above formula applies directly: \[\begin{align} \text{Pr}(X=0) &= b(0;5,0.25) = \binom{5}{0}(0.25)^0(0.75)^5 \approx 0.237\\ \text{Pr}(X=1) &= b(1;5,0.25) = \binom{5}{1}(0.25)^1(0.75)^4 \approx 0.396\\ \text{Pr}(X=2) &= b(2;5,0.25) = \binom{5}{2}(0.25)^2(0.75)^3 \approx 0.263\\ \text{Pr}(X=3) &= b(3;5,0.25) = \binom{5}{3}(0.25)^3(0.75)^2 \approx 0.088\\ \text{Pr}(X=4) &= b(4;5,0.25) = \binom{5}{4}(0.25)^4(0.75)^1 \approx 0.015\\ \text{Pr}(X=5) &= b(5;5,0.25) = \binom{5}{5}(0.25)^5(0.75)^0 \approx 0.001. \end{align}\] It's worth noting that the most likely result is to flip one head, which is explored further below when discussing the mode of the distribution. \(_\square\)

This can be represented pictorially, as in the following table:

The binomial distribution \(b(5,0.25)\)

You have an (extremely) biased coin that shows heads with probability 99% and tails with probability 1%. To test the coin, you tossed it 100 times.

What is the approximate probability that heads showed up exactly \( 99 \) times?

A fair coin is flipped 10 times. What is the probability that it lands on heads the same number of times that it lands on tails?

Give your answer to three decimal places.

There are several important values that give information about a particular probability distribution. The most important are as follows:

• The mean , or expected value , of a distribution gives useful information about what average one would expect from a large number of repeated trials.
• The median of a distribution is another measure of central tendency, useful when the distribution contains outliers (i.e. particularly large/small values) that make the mean misleading.
• The mode of a distribution is the value that has the highest probability of occurring.
• The variance of a distribution measures how "spread out" the data is. Related is the standard deviation , the square root of the variance, useful due to being in the same units as the data.

Three of these values--the mean, mode, and variance--are generally calculable for a binomial distribution. The median, however, is not generally determined.

The mean of a binomial distribution is intuitive:

The mean of \(b(n,p)\) is \(np.\)

In other words, if an unfair coin that flips heads with probability \(p\) is flipped \(n\) times, the expected result would be \(np\) heads.

Let \(X_1, X_2, \ldots, X_n\) be random variables representing the Bernoulli trial with probability \(p\) of success. Then \(X = X_1 + X_2 + \cdots + X_n\), by definition. By linearity of expectation , \[E[X]=E[X_1+X_2+\cdots+X_n]=E[X_1]+E[X_2]+\cdots+E[X_n]=\underbrace{p+p+\cdots+p}_{n\text{ times}}=np.\ _\square\]

You have an (extremely) biased coin that shows heads with 99% probability and tails with 1% probability.

If you toss it 100 times, what is the expected number of times heads will come up?

This problem is part of the set Extremely Biased Coins.

A similar strategy can be used to determine the variance:

The variance of \(b(n,p)\) is \(np(1-p)\).

Since variance is additive, a similar proof to the above can be used: \[ \begin{align*} \text{Var}[X] &= \text{Var}(X_1 + X_2 + \cdots + X_n) \\ &= \text{Var}(X_1) + \text{Var}(X_2) + \cdots + \text{Var}(X_n) \\ &= \underbrace{p(1-p)+p(1-p)+\cdots+p(1-p)}_{n\text{ times}} \\ &= np(1-p) \end{align*} \] since the variance of a single Bernoulli trial is \(p(1-p)\). \(_\square\)

The mode, however, is slightly more complicated. In most cases the mode is \(\lfloor (n+1)p \rfloor\), but if \((n+1)p\) is an integer, both \((n+1)p\) and \((n+1)p-1\) are modes. Additionally, in the trivial cases of \(p=0\) and \(p=1\), the modes are 0 and \(n,\) respectively.

The mode of \(b(n,p)\) is

\[ \text{mode} = \begin{cases} 0 & \text{if } p = 0 \\ n & \text{if } p = 1 \\ (n+1)\,p\ \text{ and }\ (n+1)\,p - 1 &\text{if }(n+1)p\in\mathbb{Z} \\ \big\lfloor (n+1)\,p\big\rfloor & \text{if }(n+1)p\text{ is 0 or a non-integer}. \end{cases} \]

Daniel has a weighted coin that flips heads \(\frac{2}{5}\) of the time and tails \(\frac{3}{5}\) of the time. If he flips it \(9\) times, the probability that it will show heads exactly \(n\) times is greater than or equal to the probability that it will show heads exactly \(k\) times, for all \(k=0, 1,\dots, 9, k\ne n\).

If the probability that the coin will show heads exactly \(n\) times in \(9\) flips is \(\frac{p}{q}\) for positive coprime integers \(p\) and \(q\), then find the last three digits of \(p\).

The binomial distribution is applicable to most situations in which a specific target result is known, by designating the target as "success" and anything other than the target as "failure." Here is an example:

A die is rolled 3 times. What is the probability that no sixes occur? In this binomial experiment, rolling anything other than a 6 is a success and rolling a 6 is failure. Since there are three trials, the desired probability is \[b\left(3;3,\frac{5}{6}\right)=\binom{3}{3}\left(\frac{5}{6}\right)^3\left(\frac{1}{6}\right)^0 \approx .579.\] This could also be done by designating rolling a 6 as a success, and rolling anything else as failure. Then the desired probability would be \[b\left(0;3,\frac{1}{6}\right)=\binom{3}{0}\left(\frac{1}{6}\right)^0\left(\frac{5}{6}\right)^3 \approx .579\] just as before. \(_\square\)

The binomial distribution is also useful in analyzing a range of potential results, rather than just the probability of a specific one:

A manufacturer of widgets knows that 20% of the widgets he produces are defective. If he produces 10 widgets per day, what is the probability that at most two of them are defective? In this binomial experiment, manufacturing a working widget is a success and manufacturing a defective widget is a failure. The manufacturer needs at least 8 successes, making the probability \[ \begin{align*} b(8;10,0.8)+b(9;10,0.8)+b(10;10,0.8) &=\binom{10}{8}(0.8)^8(0.2)^2+\binom{10}{9}(0.8)^9(0.2)^1+\binom{10}{10}(0.8)^{10} \\\\ &\approx 0.678. \ _\square \end{align*} \]

This example also illustrates an important clash with intuition: generally, one would expect that an 80% success rate is appropriate when requiring 8 of 10 widgets to not be defective. However, the above calculation shows that an 80% success rate only results in at least 8 successes less than 68% of the time!

This calculation is especially important for a related reason: since the manufacturer knows his error rate and his quota, he can use the binomial distribution to determine how many widgets he must produce in order to earn a sufficiently high probability of meeting his quota of non-defective widgets.

Related to the final note of the last section, the binomial test is a method of testing for statistical significance . Most commonly, it is used to reject the null hypothesis of uniformity; for example, it can be used to show that a coin or die is unfair. In other words, it is used to show that the given data is unlikely under the assumption of fairness, so that the assumption is likely false.

A coin is flipped 100 times, and the results are 61 heads and 39 tails. Is the coin fair? The null hypothesis is that the coin is fair, in which case the probability of flipping at least 61 heads is \[\sum_{i=61}^{100}b(i;100,0.5) = \sum_{i=61}^{100}\binom{100}{i}(0.5)^{100} \approx 0.0176,\] or \(1.76\%\). Determining whether this result is statistically significant depends on the desired confidence level; this would be enough to reject the null hypothesis at the 5% level, but not the 1% one. As the most commonly used confidence level is the 5% one, this would generally be considered sufficient to conclude that the coin is unfair. \(_\square\)

• Geometric Distribution
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Binomial distribution

by Marco Taboga , PhD

The binomial distribution is a univariate discrete distribution used to model the number of favorable outcomes obtained in a repeated experiment.

Table of contents

How the distribution is used

What you need to know, relation to the bernoulli distribution, expected value, moment generating function, characteristic function, distribution function, solved exercises.

Consider an experiment having two possible outcomes: either success or failure.

Suppose that the experiment is repeated several times and the repetitions are independent of each other.

The distribution of the number of experiments in which the outcome turns out to be a success is called binomial distribution.

Chart of binomial distribution with interactive calculator

A binomial distribution can be seen as a sum of mutually independent Bernoulli random variables that take value 1 in case of success of the experiment and value 0 otherwise.

This connection between the binomial and Bernoulli distributions will be illustrated in detail in the remainder of this lecture and will be used to prove several properties of the binomial distribution.

Before proceeding, you are advised to study the lecture on the Bernoulli distribution .

The binomial distribution is characterized as follows.

The binomial distribution is intimately related to the Bernoulli distribution. The following propositions show how.

binocdf(x,n,p)

returns the value of the distribution function at the point x when the parameters of the distribution are n and p .

You can also use the calculator at the top of this page.

Below you can find some exercises with explained solutions.

How to cite

Please cite as:

Taboga, Marco (2021). "Binomial distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/probability-distributions/binomial-distribution.

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4.3 The Binomial Distribution

We have seen how to deal with general discrete random variables , but there are also special cases of DRVs.  If we can identify them, they can provide us some insight and shortcuts.  The first of these is the Binomial Distribution.

The Binomial Setting

There are three characteristics of a binomial experiment .

• There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
• There are only two possible outcomes, called “success” and “failure,” for each trial. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial. p + q = 1.
• The n trials are independent and are repeated using identical conditions. Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, p , of a success and probability, q , of a failure remain the same.

For example: At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A “success” could be defined as an individual who withdrew. The random variable X = the number of students who withdraw from the randomly selected elementary physics class.

Any experiment that has characteristics two and three and where n = 1 is called a Bernoulli trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes place when the number of successes is counted in one or more Bernoulli trials.

For example, randomly guessing at a true-false statistics question has only two outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true-false question with probability p = 0.6. Then, q = 0.4. This means that for every true-false statistics question Joe answers, his probability of success ( p = 0.6) and his probability of failure ( q = 0.4) remain the same.  This situation meets the Binomial requirements.

Approximately 70% of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time? Students are selected randomly.

a. This is a binomial problem because there is only a success or a __________, there are a fixed number of trials, and the probability of a success is 0.70 for each trial.

b. If we are interested in the number of students who do their homework on time, then how do we define X ?

c. What values does x take on?

d. What is a “failure,” in words?

e. If p + q = 1, then what is q ?

f. The words “at least” translate as what kind of inequality for the probability question P ( x ____ 40).

Sixty-five percent of people pass the state driver’s exam on the first try. A group of 50 individuals who have taken the driver’s exam is randomly selected. Can we use the binomial here?

Notation for the Binomial

The outcomes of a binomial experiment fit a binomial probability distribution . The random variable X counts the number of successes obtained in the n independent trials.

X ~ B ( n , p )

Read this as “ X is a random variable with a binomial distribution.” The parameters are n and p:   n = number of trials, p = probability of a success on each trial.

Since the Binomial counts the number of successes, x, in n trials, the range of vaules for a binomial random variable could be anything from 0 to n (x=0,1,2…, n).

Binomial Probability Function

Once we have decided we can use the binomial for a given situation, we can use the binomial probability function to find the probability of a specific number of successes, P(X=x).  The binomial PMF is made up of two parts:

First, we need to find out how many different ways we can get x successes in n trials.  To do this we can use the “Choose” function, also called the binomial coefficient, written as:

Note: The the ! mark is the factorial operator.

The next part gives us the probability of a single one of those ways to get x successes in n trials.  We can do this by using our independent multiplication rule.   We multiply the probability of success (p) raised to the number of successes (x) by the probability of failure (q=1-p) raised to the number of failures (n-x).

p x q (n-x)

Since we know each of these ways are equally likely and how many ways are possible we can now put the two pieces together. We multiply the probability of one way by how many we have to give us our overall probability of x successes in n trials.

Unfortunately the binomial does not have a nice form of CDF , but it is simply the sum of PDFs up until that point. Consider the following example to demonstrate this point.

It has been stated that about 41% of adult workers have a high school diploma but do not pursue any further education. 20 adult workers are randomly selected.

Let X = the number of workers who have a high school diploma but do not pursue any further education.

X takes on the values 0, 1, 2, …, 20 where n = 20, p = 0.41, and q = 1 – 0.41 = 0.59. X ~ B (20, 0.41)

The  y -axis contains the probability of  x , where  X  = the number of workers who have only a high school diploma.

The graph of  X  ~  B (20, 0.41) is as follows:

Find the probability that:

(a) Exactly 12 of them have a high school diploma

(b) At most 12 of them have a high school diploma but do not pursue any further education. How many adult workers do you expect to have a high school diploma but do not pursue any further education?

About 32% of students participate in a community volunteer program outside of school. If 30 students are selected at random, find:

(a) The probability that exactly 14 of them participate in a community volunteer program outside of school.  First try plugging in to the binomial formula by hand, then check yourself with technology.

(b) The probability that exactly 14 of them participate in a community volunteer program outside of school. Rely on technology for this cumulative probability.

Measures of the Binomial Distribution

In the 2013 Jerry’s Artarama art supplies catalog, there are 560 pages. Eight of the pages feature signature artists. Suppose we randomly sample 100 pages. Let X = the number of pages that feature signature artists.

• What values does x take on?
• What is the probability distribution? Find the following probabilities

2a. the probability that two pages feature signature artists.

2b. the probability that at most six pages feature signature artists

2c. the probability that more than three pages feature signature artists.

3. Using the formulas, calculate the (3a) mean and (3b) standard deviation.

According to a Gallup poll, 60% of American adults prefer saving over spending. Let X = the number of American adults out of a random sample of 50 who prefer saving to spending.

• What is the probability distribution for X ?
• the probability that 25 adults in the sample prefer saving over spending
• the probability that at most 20 adults prefer saving
• the probability that more than 30 adults prefer saving
• Using the formulas, calculate the (i) mean and (ii) standard deviation of X .

Image References

A random variable that produces discrete data

A random variable that counts the number of successes in a fixed number (n) of independent Bernoulli trials each with probability of a success (p)

The occurrence of one event has no effect on the probability of the occurrence of another event

An experiment with the following characteristics:

- There are only two possible outcomes called “success” and “failure” for each trial - The probability (p) of a success is the same for any trial (so the probability q = 1 − p of a failure is the same for any trial)

A function that gives the probability that a discrete random variable is exactly equal to some value (x)

A function that gives the probability that a random variable takes a value less than or equal to x

Significant Statistics Copyright © 2020 by John Morgan Russell, OpenStaxCollege, OpenIntro is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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Module 4: Discrete Random Variables

What are binomial experiments, learning outcomes.

• Describe the three characteristics of a binomial experiment

There are three characteristics of a binomial experiment .

• There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter [latex]n[/latex] denotes the number of trials.
• There are only two possible outcomes, called “success” and “failure,” for each trial. The letter [latex]p[/latex]   denotes the probability of a success on one trial, and [latex]q[/latex] denotes the probability of a failure on one trial. [latex]p+q=1[/latex].
• The [latex]n[/latex] trials are independent and are repeated using identical conditions. Because the [latex]n[/latex] trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, [latex]p[/latex] of a success and probability, [latex]q[/latex], of a failure remain the same. For example, randomly guessing at a true-false statistics question has only two outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true-false question with probability [latex]p=0.6[/latex]. Then, [latex]q=0.4[/latex]. This means that for every true-false statistics question Joe answers, his probability of success [latex](p=0.6)[/latex] and his probability of failure [latex](q=0.4)[/latex] remain the same.

The outcomes of a binomial experiment fit a binomial probability distribution . The random variable [latex]X=[/latex] the number of successes obtained in the [latex]n[/latex] independent trials.

The mean, [latex]\mu[/latex], and variance, [latex]\sigma^{2}[/latex], for the binomial probability distribution are [latex]\mu=np[/latex] and [latex]\sigma^{2}=npq[/latex]. The standard deviation, [latex]\sigma[/latex], is then [latex]\sigma=\sqrt{{{n}{p}{q}}}[/latex].

Any experiment that has characteristics two and three and where [latex]n=1[/latex] is called a Bernoulli Trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes place when the number of successes is counted in one or more Bernoulli Trials.

At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A “success” could be defined as an individual who withdrew. The random variable [latex]X=[/latex] the number of students who withdraw from the randomly selected elementary physics class.

The state health board is concerned about the amount of fruit available in school lunches. Forty-eight percent of schools in the state offer fruit in their lunches every day. This implies that 52% do not. What would a “success” be in this case?

Suppose you play a game that you can only either win or lose. The probability that you win any game is 55%, and the probability that you lose is 45%. Each game you play is independent. If you play the game 20 times, write the function that describes the probability that you win 15 of the 20 times.

A trainer is teaching a dolphin to do tricks. The probability that the dolphin successfully performs the trick is 35%, and the probability that the dolphin does not successfully perform the trick is 65%. Out of 20 attempts, you want to find the probability that the dolphin succeeds 12 times. State the probability question mathematically.

A fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than ten heads? Let [latex]X=[/latex] the number of heads in 15 flips of the fair coin. [latex]X[/latex] takes on the values 0, 1, 2, 3, …, 15. Since the coin is fair, [latex]p=0.5[/latex] and [latex]q=0.5[/latex]. The number of trials is [latex]n=15[/latex]. State the probability question mathematically.

A fair, six-sided die is rolled ten times. Each roll is independent. You want to find the probability of rolling a one more than three times. State the probability question mathematically.

Approximately 70% of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time? Students are selected randomly.

• This is a binomial problem because there is only a success or a __________, there are a fixed number of trials, and the probability of a success is 0.70 for each trial.
• If we are interested in the number of students who do their homework on time, then how do we define [latex]X[/latex]?
• What values does [latex]x[/latex] take on?
• What is a “failure,” in words?
• If [latex]p+q=1[/latex], then what is [latex]q[/latex]?
• The words “at least” translate as what kind of inequality for the probability question [latex]P(x\geq40)[/latex].
• [latex]X=[/latex] the number of statistics students who do their homework on time
• 0, 1, 2, …, 50
• Failure is defined as a student who does not complete his or her homework on time. The probability of a success is [latex]p=0.70[/latex]. The number of trials is [latex]n=50[/latex].
• [latex]q=0.30[/latex]
• greater than or equal to (≥)The probability question is [latex]P(x\geq40)[/latex].

Sixty-five percent of people pass the state driver’s exam on the first try. A group of 50 individuals who have taken the driver’s exam is randomly selected. Give two reasons why this is a binomial problem.

• Binomial Distribution. Provided by : OpenStax. Located at : https://openstax.org/books/introductory-statistics/pages/4-3-binomial-distribution . License : CC BY: Attribution . License Terms : Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction
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Binomial Probability Calculator

Use the Binomial Calculator to compute individual and cumulative binomial probabilities. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems .

To learn more about the binomial distribution, go to Stat Trek's tutorial on the binomial distribution .

• Enter a value in each of the first three text boxes (the unshaded boxes).
• Click the Calculate button to compute binomial and cumulative probabilities.

Frequently-Asked Questions

Instructions: To find the answer to a frequently-asked question, simply click on the question.

What is a binomial experiment?

A binomial experiment has the following characteristics:

• The experiment involves repeated trials.
• Each trial has only two possible outcomes - a success or a failure.
• The probability that any trial will result in success is constant.
• All of the trials in the experiment are independent.

A series of coin tosses is a perfect example of a binomial experiment. Suppose we toss a coin three times. Each coin flip represents a trial, so this experiment would have 3 trials. Each coin flip also has only two possible outcomes - a Head or a Tail. We could call a Head a success; and a Tail, a failure. The probability of a success on any given coin flip would be constant (i.e., 50%). And finally, the outcome on any coin flip is not affected by previous or succeeding coin flips; so the trials in the experiment are independent.

What is a binomial distribution?

A binomial distribution is a probability distribution . It refers to the probabilities associated with the number of successes in a binomial experiment .

For example, suppose we toss a coin three times and suppose we define Heads as a success. This binomial experiment has four possible outcomes: 0 Heads, 1 Head, 2 Heads, or 3 Heads. The probabilities associated with each possible outcome are an example of a binomial distribution , as shown below.

What is the number of trials?

The number of trials refers to the number of replications in a binomial experiment.

Suppose that we conduct the following binomial experiment. We flip a coin and count the number of Heads. We classify Heads as success; tails, as failure. If we flip the coin 3 times, then 3 is the number of trials. If we flip it 20 times, then 20 is the number of trials.

Note: Each trial results in a success or a failure. So the number of trials in a binomial experiment is equal to the number of successes plus the number of failures.

What is the number of successes?

Each trial in a binomial experiment can have one of two outcomes. The experimenter classifies one outcome as a success; and the other, as a failure. The number of successes in a binomial experient is the number of trials that result in an outcome classified as a success.

What is the probability of success on a single trial?

In a binomial experiment, the probability of success on any individual trial is constant. For example, the probability of getting Heads on a single coin flip is always 0.50. If "getting Heads" is defined as success, the probability of success on a single trial would be 0.50.

What is the binomial probability?

A binomial probability refers to the probability of getting EXACTLY r successes in a specific number of trials. For instance, we might ask: What is the probability of getting EXACTLY 2 Heads in 3 coin tosses. That probability (0.375) would be an example of a binomial probability.

In a binomial experiment, the probability that the experiment results in exactly x successes is indicated by the following notation: P(X=x);

What is the cumulative binomial probability?

Cumulative binomial probability refers to the probability that the value of a binomial random variable falls within a specified range.

The probability of getting AT MOST 2 Heads in 3 coin tosses is an example of a cumulative probability. It is equal to the probability of getting 0 heads (0.125) plus the probability of getting 1 head (0.375) plus the probability of getting 2 heads (0.375). Thus, the cumulative probability of getting AT MOST 2 Heads in 3 coin tosses is equal to 0.875.

Notation associated with cumulative binomial probability is best explained through illustration. The probability of getting FEWER THAN 2 successes is indicated by P(X<2); the probability of getting AT MOST 2 successes is indicated by P(X≤2); the probability of getting AT LEAST 2 successes is indicated by P(X≥2); the probability of getting MORE THAN 2 successes is indicated by P(X>2).

Sample Problem

• The probability of success (i.e., getting a Head) on any single trial is 0.5.
• The number of trials is 12.
• The number of successes is 7 (since we define getting a Head as success).

Therefore, we plug those numbers into the Binomial Calculator and hit the Calculate button.

The calculator reports that the binomial probability is 0.193. That is the probability of getting EXACTLY 7 Heads in 12 coin tosses. (The calculator also reports the cumulative probabilities. For example, the probability of getting AT MOST 7 heads in 12 coin tosses is a cumulative probability equal to 0.806.)

• The probability of success for any individual student is 0.6.
• The number of trials is 3 (because we have 3 students).
• The number of successes is 2.

The calculator reports that the probability that two or fewer of these three students will graduate is 0.784.

Oops! Something went wrong.

7.10 The Binomial Distribution

Learning objectives.

After completing this section, you should be able to:

• Identify binomial experiments.
• Use the binomial distribution to analyze binomial experiments.

It’s time for the World Series, which determines the champion for this season in Major League Baseball. The scrappy Los Angeles Angels are facing the powerhouse Cincinnati Reds. Computer models put the chances of the Reds winning any single game against the Angels at about 65%. The World Series, as its name implies, isn’t just one game, though: it’s what’s known as a “best-of-seven” contest: the teams play each other repeatedly until one team wins 4 games (which could take up to 7 games total, if each team wins three of the first 6 matchups). If the Reds truly have a 65% chance of winning a single game, then the probability that they win the series should be greater than 65%. Exactly how much bigger?

If you have the patience for it, you could use a tree diagram like we used in Example 7.33 to trace out all of the possible outcomes, find all the related probabilities, and add up the ones that result in the Reds winning the series. Such a tree diagram would have 2 7 = 128 2 7 = 128 final nodes, though, so the calculations would be very tedious. Fortunately, we have tools at our disposal that allow us to find these probabilities very quickly. This section will introduce those tools and explain their use.

Binomial Experiments

The tools of this section apply to multistage experiments that satisfy some pretty specific criteria. Before we move on to the analysis, we need to introduce and explain those criteria so that we can recognize experiments that fall into this category. Experiments that satisfy each of these criteria are called binomial experiments . A binomial experiment is an experiment with a fixed number of repeated independent binomial trials, where each trial has the same probability of success.

Repeated Binomial Trials

The first criterion involves the structure of the stages. Each stage of the experiment should be a replication of every other stage; we call these replications trials . An example of this is flipping a coin 10 times; each of the ten flips is a trial, and they all occur under the same conditions as every other. Further, each trial must have only two possible outcomes. These two outcomes are typically labeled “success” and “failure,” even if there is not a positive or negative connotation associated with those outcomes. Experiments with more than two outcomes in their sample spaces are sometimes reconsidered in a way that forces just two outcomes; all we need to do is completely divide the sample space into two parts that we can label “success” and “failure.” For example, your grade on an exam might be recorded as A, B, C, D, or F, but we could instead think of the grades A, B, C, and D as “success” and a grade of F as “failure.” Trials with only two outcomes are called binomial trials (the word binomial derives from Latin and Greek roots that mean “two parts”).

Independent Trials

The next criterion that we’ll be looking for is independence of trials. Back in Tree Diagrams, Tables, and Outcomes , we said that two stages of an experiment are independent if the outcome of one stage doesn’t affect the other stage. Independence is necessary for the experiments we want to analyze in this section.

Fixed Number of Trials

Next, we require that the number of trials in the experiment be decided before the experiment begins. For example, we might say “flip a coin 10 times.” The number of trials there is fixed at 10. However, if we say “flip a coin until you get 5 heads,” then the number of trials could be as low as 5, but theoretically it could be 50 or a 100 (or more)! We can’t apply the tools from this section in cases where the number of trials is indeterminate.

Constant Probability

The next criterion needed for binomial experiments is related to the independence of the trials. We must make sure that the probability of success in each trial is the same as the probability of success in every other trial.

Example 7.34

Identifying binomial experiments.

Decide whether each of the following is a binomial experiment. For those that aren’t, identify which criterion or criteria are violated.

• You roll a standard 6-sided die 10 times and write down the number that appears each time.
• You roll a standard 6-sided die 10 times and write down whether the die shows a 6 or not.
• You roll a standard 6-sided die until you get a 6.
• You roll a standard 6-sided die 10 times. On the first roll, we define “success” as rolling a 4 or greater. After the first roll, we define “success” as rolling a number greater than the result of the previous roll.
• Since we’re noting 1 of 6 possible outcomes, the trials are not binomial. So, this isn’t a binomial experiment.
• We have 2 possible outcomes (“6” and “not 6”), the trials are independent, the probability of success is the same every time, and the number of trials is fixed. This is a binomial experiment.
• Since the number of trials isn’t fixed (we don’t know if we’ll get our first 6 after 1 roll or 20 rolls or somewhere in between), this isn’t a binomial experiment.
• Here, the probability of success might change with every roll (on the first roll, that probability is 1 2 1 2 ; if the first roll is a 6, the probability of success on the next roll is zero). So, this is not a binomial experiment.

Your Turn 7.34

The binomial formula.

If we flip a coin 100 times, you might expect the number of heads to be around 50, but you probably wouldn’t be surprised if the actual number of heads was 47 or 52. What is the probability that the number of heads is exactly 50? Or falls between 45 and 55? It seems unlikely that we would get more than 70 heads. Exactly how unlikely is that?

Each of these questions is a question about the number of successes in a binomial experiment (flip a coin 100 times, “success” is flipping heads). We could theoretically use the techniques we’ve seen in earlier sections to answer each of these, but the number of calculations we’d have to do is astronomical; just building the tree diagram that represents this situation is more than we could complete in a lifetime; it would have 2 100 ≈ 1.3 × 10 30 2 100 ≈ 1.3 × 10 30 final nodes! To put that number in perspective, if we could draw 1,000 dots every second, and we started at the moment of the Big Bang, we’d currently be about 0.00000003% of the way to drawing out those final nodes. Luckily, there’s a shortcut called the Binomial Formula that allows us to get around doing all those calculations!

Binomial Formula: Suppose we have a binomial experiment with n n trials and the probability of success in each trial is p p . Then:

We can use this formula to answer one of our questions about 100 coin flips. What is the probability of flipping exactly 50 heads? In this case, n = 100 n = 100 , p = 1 2 p = 1 2 , and a = 50 a = 50 , so P ( flip 50 heads ) = C 100 50 × ( 1 2 ) 50 × ( 1 − 1 2 ) 100 − 50 P ( flip 50 heads ) = C 100 50 × ( 1 2 ) 50 × ( 1 − 1 2 ) 100 − 50 . Unfortunately, many calculators will balk at this calculation; that first factor ( 100 C 50 100 C 50 ) is an enormous number, and the other two factors are very close to zero. Even if your calculator can handle numbers that large or small, the arithmetic can create serious errors in rounding off.

Luckily, spreadsheet programs have alternate methods for doing this calculation. In Google Sheets, we can use the BINOMDIST function to do this calculation for us. Open up a new sheet, click in any empty cell, and type “=BINOMDIST(50,100,0.5,FALSE)” followed by the Enter key. The cell will display the probability we seek; it’s about 8%. Let’s break down the syntax of that function in Google Sheets: enter “=BINOMDIST( a a , n n , p p , FALSE)” to find the probability of a a successes in n n trials with probability of success p p .

Example 7.35

Using the binomial formula.

• Find the probability of rolling a standard 6-sided die 4 times and getting exactly one 6 without using technology .
• Find the probability of rolling a standard 6-sided die 60 times and getting exactly ten 6s using technology.
• Find the probability of rolling a standard 6-sided die 60 times and getting exactly eight 6s using technology.
• We’ll apply the Binomial Formula, where n = 4 n = 4 , a = 1 a = 1 , and p = 1 6 p = 1 6 : P ( rolling one 6 ) = C 1 4 × ( 1 6 ) 1 × ( 5 6 ) 4 − 1 = 4 ! 1 ! ( 4 − 1 ) ! × 1 6 × ( 5 6 ) 3 = 4 × 1 6 × 5 3 6 3 = 4 × 5 3 6 4 = 500 1,296 . P ( rolling one 6 ) = C 1 4 × ( 1 6 ) 1 × ( 5 6 ) 4 − 1 = 4 ! 1 ! ( 4 − 1 ) ! × 1 6 × ( 5 6 ) 3 = 4 × 1 6 × 5 3 6 3 = 4 × 5 3 6 4 = 500 1,296 .
• Here, n = 60 n = 60 , a = 10 a = 10 , and p = 1 6 p = 1 6 . In Google Sheets, we’ll enter “=BINOMDIST(10, 60, 1/6, FALSE)” to get our result: 0.137.
• This experiment is the same as in Exercise 2 of this example; we’re simply changing the number of successes from 10 to 8. Making that change in the formula in Google Sheets, we get the probability 0.116.

Your Turn 7.35

The binomial distribution.

If we are interested in the probability of more than just a single outcome in a binomial experiment, it’s helpful to think of the Binomial Formula as a function, whose input is the number of successes and whose output is the probability of observing that many successes. Generally, for a small number of trials, we’ll give that function in table form, with a complete list of the possible outcomes in one column and the probability in the other.

For example, suppose Kristen is practicing her basketball free throws. Assume Kristen always makes 82% of those shots. If she attempts 5 free throws, then the Binomial Formula gives us these probabilities:

A table that lists all possible outcomes of an experiment along with the probabilities of those outcomes is an example of a probability density function (PDF). A PDF may also be a formula that you can use to find the probability of any outcome of an experiment.

Because they refer to the same thing, some sources will refer to the Binomial Formula as the Binomial PDF.

If we want to know the probability of a range of outcomes, we could add up the corresponding probabilities. Going back to Kristen’s free throws, we can find the probability that she makes 3 or fewer of her 5 attempts by adding up the probabilities associated with the corresponding outcomes (in this case: 0, 1, 2, or 3):

The probability that the outcome of an experiment is less than or equal to a given number is called a cumulative probability . A table of the cumulative probabilities of all possible outcomes of an experiment is an example of a cumulative distribution function (CDF). A CDF may also be a formula that you can use to find those cumulative probabilities.

Cumulative probabilities are always associated with events that are defined using ≤ ≤ . If other inequalities are used to define the event, we must restate the definition so that it uses the correct inequality.

Here are the PDF and CDF for Kristen’s free throws:

Google Sheets can also compute cumulative probabilities for us; all we need to do is change the “FALSE” in the formulas we used before to "TRUE."

Example 7.36

Using the binomial cdf.

Suppose we are about to flip a fair coin 50 times. Let H H represent the number of heads that result from those flips. Use technology to find the following:

• P ( H ≤ 22 ) P ( H ≤ 22 )
• P ( H < 26 ) P ( H < 26 )
• P ( H > 28 ) P ( H > 28 )
• P ( H ≥ 20 ) P ( H ≥ 20 )
• P ( 20 < H < 25 ) P ( 20 < H < 25 )
• The event here is defined by H ≤ 22 H ≤ 22 , which is the inequality we need to have if we want to use the Binomial CDF. In Google Sheets, we’ll enter “=BINOMDIST(22, 50, 0.5, TRUE)” to get our answer: 0.2399.
• This event uses the wrong inequality, so we need to do some preliminary work. If H < 26 H < 26 , that means H ≤ 25 H ≤ 25 (because H H has to be a whole number). So, we’ll enter “=BINOMDIST(25, 50, 0.5, TRUE)” to find P ( H < 26 ) = P ( H ≤ 25 ) = 0.5561 P ( H < 26 ) = P ( H ≤ 25 ) = 0.5561 .
• The inequality associated with this event is pointing in the wrong direction. If E E is the event H > 28 H > 28 , that means that E E contains the outcomes {29, 30, 31, 32, 33, …}. Thus, E ′ E ′ must contain the outcomes {…, 25, 26, 27, 28}. In other words, E ′ E ′ is defined by H ≤ 28 H ≤ 28 . Since it uses ≤ ≤ , we can find P ( E ′ ) P ( E ′ ) using “=BINOMDIST(28, 50, 0.5, TRUE)”: 0.8389 So, using the formula for probabilities of complements, we have P ( E ) = 1 − P ( E ′ ) = 1 − 0.8389 = 0.1611. P ( E ) = 1 − P ( E ′ ) = 1 − 0.8389 = 0.1611.
• As in part 3, this inequality is pointing in the wrong direction. If F F is the event H ≥ 20 H ≥ 20 , then F F contains the outcomes {20, 21, 22, 23, …}. That means F ′ F ′ contains the outcomes {…, 16, 17, 18, 19}, and so F ′ F ′ is defined by H ≤ 19 H ≤ 19 . So, we can find P ( F ′ ) P ( F ′ ) using “=BINOMDIST(19, 50, 0.5, TRUE)”: 0.0595. Finally, using the formula for probabilities of complements, we get: P ( F ) = 1 − P ( F ′ ) = 1 − 0.0595 = 0.9405. P ( F ) = 1 − P ( F ′ ) = 1 − 0.0595 = 0.9405.
• If 20 < H < 25 20 < H < 25 , that means we are interested in the outcomes {21, 22, 23, 24}. This doesn’t look like any of the previous situations, but there is a way to find this probability using the Binomial CDF. We need to put everything in terms of “less than or equal to,” so we’ll first note that all of our outcomes are less than or equal to 24. But we don’t want to include values that are less than or equal to 20. So, we have three events: let I I be the event defined by 20 < H < 25 20 < H < 25 (note that we’re trying to find P ( I ) P ( I ) ). Let J J be defined by H ≤ 24 H ≤ 24 , and let K K be defined by H ≤ 20 H ≤ 20 . Of these three events, J J contains the most outcomes. If J J occurs, then either K K or I I must have occurred. Moreover, K K and I I are mutually exclusive. Thus, P ( J ) = P ( K ) + P ( I ) P ( J ) = P ( K ) + P ( I ) , by the Addition Rule. Solving for the probability that we want, we get P ( I ) = P ( J ) − P ( K ) = P ( H ≤ 24 ) − P ( H ≤ 20 ) = 0.44386 − 0.10132 = 0.34254. P ( I ) = P ( J ) − P ( K ) = P ( H ≤ 24 ) − P ( H ≤ 20 ) = 0.44386 − 0.10132 = 0.34254.

Your Turn 7.36

Finally, we can answer the question posed at the beginning of this section . Remember that the Reds are facing the Angels in the World Series, which is won by the team who is first to win 4 games. The Reds have a 65% chance to win any game against the Angels. So, what is the probability that the Reds win the World Series? At first glance, this is not a binomial experiment: The number of games played is not fixed, since the series ends as soon as one team wins 4 games. However, we can extend this situation to a binomial experiment: Let’s assume that 7 games are always played in the World Series, and the winner is the team who wins more games. In a way, this is what happens in reality; it’s as though the first team to lose 4 games (and thus cannot win more than the other team) forfeits the rest of their games. So, we can treat the actual World Series as a binomial experiment with seven trials. If W W is the number of games won by the Reds, the probability that the Reds win the World Series is P ( W ≥ 4 ) P ( W ≥ 4 ) . Using the techniques from the last example, we get P ( Reds win the series ) = 0.8002 P ( Reds win the series ) = 0.8002 .

People in Mathematics

Abraham de moivre.

Abraham de Moivre was born in 1667 in France to a Protestant family. Though he was educated in Catholic schools, he remained true to his faith; in 1687, he fled with his brother to London to escape persecution under the reign of King Louis XIV. Once he arrived in England, he supported himself as a freelance math tutor while he conducted his own research. Among his interests was probability; in 1711, he published the first edition of The Doctrine of Chances: A Method of Calculating the Probabilities of Events in Play . This book was the second textbook on probability (after Cardano’s Liber de ludo aleae ). De Moivre discovered an important connection between the binomial distribution and the normal distribution (an important concept in statistics; we’ll explore that distribution and its connection to the binomial distribution in Chapter 8). De Moivre also discovered some properties of a new probability distribution that later became known as the Poisson distribution.

Check Your Understanding

Section 7.10 exercises.

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Binomial Experiment: Rules, Examples, Steps

What is a binomial experiment.

A binomial experiment is an experiment where you have a fixed number of independent trials with only have two outcomes. For example, the outcome might involve a yes or no answer. If you toss a coin you might ask yourself “Will I get a heads?” and the answer is either yes or no. That’s the basic idea, but in order to call an experiment a binomial experiment you also have to make sure of the following rules.

• You must have a fixed number of trials . This should go without saying; if you don’t have a fixed number of trials you could be tossing that coin forever without stopping. In addition, the results from your experiment will be vastly different if you toss that coin twice (you could get two heads in a row and conclude that you will always get a heads if you toss a coin!) or if you toss it a hundred times .
• Each trial is an independent event . “Independent” means that every time you repeat the trial (i.e. tossing that coin), it’s a fresh new trial and nothing you do has an effect on each coin toss. For example, if you tossed ten coins at a time and removed the coins that landed heads down before throwing again, you’ll affect the probability, because there are fewer coins. There’s nothing wrong with that, but it would not be a binomial experiment. The fact that each trial is independent of each other leads to another important aspect of binomial experiments; the probability remains constant from trial to trial.
• There are only two outcomes. In other words, if you can phrase the experiment as a yes or no answer, then it can be a binomial experiment: Will I get a heads? Can someone find a parking space in the city? Do eggs hard boil in ten minutes?

Binomial Experiment: Examples

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• Tossing a coin a hundred times to see how many land on heads.
• Asking 100 people if they have ever been to Paris.
• Rolling two dice to see if you get a double.

Examples of experiments that are not Binomial Experiments

• Asking 100 people how much they weigh (you’ll have a hundred possible answers, not two).
• Tossing a coin until you get a heads (it could take one toss, or three, or six, so there is not a fixed number of trials). This is actually called a negative binomial experiment .

Binomial experiment: Four Steps

Determining if a question concerns a binomial experiment involves asking yourself four questions about the problem.

Example question: which of the following are binomial experiments?

• Telephone surveying a group of 200 people to ask if they voted for George Bush.
• Counting the average number of dogs seen at a veterinarian’s office daily.
• You take a survey of 50 traffic lights in a certain city, at 3 p.m., recording whether the light was red, green, or yellow at that time.
• You are at a fair, playing “pop the balloon” with 6 darts. There are 20 balloons. 10 of the balloons have a ticket inside that say “win,” and 10 have a ticket that says “lose.”

Step 1: Ask yourself: is there a fixed number of trials ?

• For question #1, the answer is yes (200).
• For question #2, the answer is no , so we’re going to discard #2 as a binomial experiment.
• For question #3, the answer is yes , there’s a fixed number of trials (the 50 traffic lights).
• For question #4, the answer is yes (your 6 darts).

Step 2: Ask yourself: Are there only 2 possible outcomes?

• For question #1, the only two possible outcomes are that they did, or they didn’t vote for Mr. Bush, so the answer is yes .
• For question #3, there are 3 possibilities: red, green, and yellow, so it’s not a binomial experiment.
• For question #4, the only possible outcomes are WIN or LOSE, so the answer is yes .

Step 3:  Ask yourself: are the outcomes independent of each other ? In other words, does the outcome of one trial (or one toss, or one question) affect another trial?

• For question #1, the answer is yes : one person saying they did or didn’t vote for Mr. Bush isn’t going to affect the next person’s response.
• For question #4, each time you toss a dart, the number of winning and losing tickets changes, which means, for example, if you win one toss, the probability of winning isn’t 10 to 10 anymore, but 9 to 10, since you already have one of the winning tickets. Since the probability is different, the trials are not independent events , so the answer is no , and question #4 is not a binomial experiment.

Step 4: Does the probability of success remain the same for each trial ?

• For question #1, the answer is yes , each person has a 50% chance of having voted for Mr. Bush.

Question #1 out of the 4 given questions was the only one that was a binomial experiment .

Check out our YouTube channel for hundreds more statistics how to videos!

Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial. Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley.

Binomial Experiments: An Explanation + Examples

Understanding binomial experiments is the first step to understanding the binomial distribution .

This tutorial defines a binomial experiment and provides several examples of experiments that are and are not  considered to be binomial experiments. 

Binomial Experiment: Definition

A  binomial experiment  is an experiment that has the following four properties:

1. The experiment consists of  n  repeated trials. The number  n  can be any amount. For example, if we flip a coin 100 times, then  n  = 100. 

2. Each trial has only two possible outcomes.  We often call outcomes either a “success” or a “failure” but a “success” is just a label for something we’re counting. For example, when we flip a coin we might call a head a “success” and a tail a “failure.”

3. The probability of success, denoted  p , is the same for each trial.  In order for an experiment to be a true binomial experiment, the probability of “success” must be the same for each trial. For example, when we flip a coin, the probability of getting heads (“success”) is always the same each time we flip the coin.

4. Each trial is independent . This simply means that the outcome of one trial does not affect the outcome of another trial. For example, suppose we flip a coin and it lands on heads. The fact that it landed on heads doesn’t change the probability that it will land on heads on the next flip. Each flip (i.e. each “trial”) is independent.

Examples of Binomial Experiments

The following experiments are all examples of binomial experiments.

Flip a coin 10 times. Record the number of times that it lands on tails.

This is a binomial experiment because it has the following four properties:

• The experiment consists of  n  repeated trials. In this case, there are 10 trials.
• Each trial has only two possible outcomes.  The coin can only land on heads or tails.
• The probability of success is the same for each trial . If we define “success” as landing on heads, then the probability of success is exactly 0.5 for each trial.
• Each trial is independent . The outcome of one coin flip does not affect the outcome of any other coin flip.

Roll a fair 6-sided die 20 times. Record the number of times that a 2 comes up.

• The experiment consists of  n  repeated trials. In this case, there are 20 trials.
• Each trial has only two possible outcomes.  If we define a 2 as a “success” then each time the die either lands on a 2 (a success) or some other number (a failure).
• The probability of success is the same for each trial . For each trial, the probability that the die lands on a 2 is 1/6. This probability does not change from one trial to the next.
• Each trial is independent . The outcome of one die roll does not affect the outcome of any other die roll.

Tyler makes 70% of his free-throw attempts. Suppose he makes 15 attempts. Record the number of baskets he makes.

• The experiment consists of  n  repeated trials. In this case, there are 15 trials.
• Each trial has only two possible outcomes. For each attempt, Tyler either makes the basket or misses it.
• The probability of success is the same for each trial . For each trial, the probability that Tyler makes the basket is 70%. This probability does not change from one trial to the next.
• Each trial is independent . The outcome of one free-throw attempt does not affect the outcome of any other free-throw attempt.

Examples that are  not Binomial Experiments

Ask 100 people how old they are .

This is not  a binomial experiment because there are more than two possible outcomes.

Roll a fair 6-sided die until a 5 comes up.

This is not  a binomial experiment because there is not a pre-defined  n  number of trials. We have no idea how many rolls it will take until a 5 comes up.

Pull 5 cards from a deck of cards. 

This is not  a binomial experiment because the outcome of one trial (e.g. pulling a certain card from the deck) affects the outcome of future trials.

A Binomial Experiment Example & Solution

The following example shows how to solve a question about a binomial experiment.

You flip a coin 10 times. What is the probability that the coin lands on heads exactly 7 times?

Whenever we’re interested in finding the probability of  n  successes in a binomial experiment, we must use the following formula:

P(exactly  k  successes) =  n C k  * p k  * (1-p) n-k

• n:  the number of trials
• k:  the number of successes
• C:  the symbol for “combination”
• p:  probability of success on a given trial

Plugging these numbers into the formula, we get:

P(7 heads) =  10 C 7  * 0.5 7  * (1-0.5) 10-7 = (120) * (.0078125) * (.125) =  0.11719 .

Thus, the probability that the coin lands on heads 7 times is  0.11719 .

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• Math Article

Binomial Distribution

In probability theory and statistics, the binomial distribution is the discrete probability distribution that gives only two possible results in an experiment, either Success or Failure . For example, if we toss a coin, there could be only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass or fail. This distribution is also called a binomial probability distribution.

There are two parameters n and p used here in a binomial distribution. The variable ‘n’ states the number of times the experiment runs and the variable ‘p’ tells the probability of any one outcome. Suppose a die is thrown randomly 10 times, then the probability of getting 2 for anyone throw is ⅙. When you throw the dice 10 times, you have a binomial distribution of n = 10 and p = ⅙. Learn the formula to calculate the two outcome distribution among multiple experiments along with solved examples here in this article.

Table of Contents:

Negative Binomial Distribution

• Mean and Variance

Binomial Distribution Vs Normal Distribution

• Solved Problems

Practice Problems

Binomial probability distribution.

In binomial probability distribution, the number of ‘Success’ in a sequence of n experiments, where each time a question is asked for yes-no, then the boolean-valued outcome is represented either with success/yes/true/one (probability p) or failure/no/false/zero (probability q = 1 − p). A single success/failure test is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process . For n = 1, i.e. a single experiment, the binomial distribution is a Bernoulli distribution . The binomial distribution is the base for the famous binomial test of statistical importance.

In probability theory and statistics, the number of successes in a series of independent and identically distributed Bernoulli trials before a particularised number of failures happens. It is termed as the negative binomial distribution. Here the number of failures is denoted by ‘r’. For instance, if we throw a dice and determine the occurrence of 1 as a failure and all non-1’s as successes. Now, if we throw a dice frequently until 1 appears the third time, i.e., r = three failures, then the probability distribution of the number of non-1s that arrived would be the negative binomial distribution.

Binomial Distribution Examples

As we already know, binomial distribution gives the possibility of a different set of outcomes. In real life, the concept is used for:

• Finding the quantity of raw and used materials while making a product.
• Taking a survey of positive and negative reviews from the public for any specific product or place.
• By using the YES/ NO survey, we can check whether the number of persons views the particular channel.
• To find the number of male and female employees in an organisation.
• The number of votes collected by a candidate in an election is counted based on 0 or 1 probability.

Also, read:

Binomial Distribution Formula

The binomial distribution formula is for any random variable X, given by;

n = the number of experiments

x = 0, 1, 2, 3, 4, …

p = Probability of Success in a single experiment

q = Probability of Failure in a single experiment = 1 – p

The binomial distribution formula can also be written in the form of n-Bernoulli trials, where n C x = n!/x!(n-x)!. Hence,

P(x:n,p) = n!/[x!(n-x)!].p x .(q) n-x

Binomial Distribution Mean and Variance

For a binomial distribution, the mean, variance and standard deviation for the given number of success are represented using the formulas

Mean, μ = np

Variance, σ 2 = npq

Standard Deviation σ= √(npq)

Where p is the probability of success

q is the probability of failure, where q = 1-p

The main difference between the binomial distribution and the normal distribution is that binomial distribution is discrete, whereas the normal distribution is continuous. It means that the binomial distribution has a finite amount of events, whereas the normal distribution has an infinite number of events. In case, if the sample size for the binomial distribution is very large, then the distribution curve for the binomial distribution is similar to the normal distribution curve.

Properties of Binomial Distribution

The properties of the binomial distribution are:

• There are two possible outcomes: true or false, success or failure, yes or no.
• There is ‘n’ number of independent trials or a fixed number of n times repeated trials.
• The probability of success or failure remains the same for each trial.
• Only the number of success is calculated out of n independent trials.
• Every trial is an independent trial, which means the outcome of one trial does not affect the outcome of another trial.

Binomial Distribution Examples And Solutions

Example 1: If a coin is tossed 5 times, find the probability of:

(a) Exactly 2 heads

(b) At least 4 heads.

(a) The repeated tossing of the coin is an example of a Bernoulli trial. According to the problem:

Number of trials: n=5

Probability of head: p= 1/2 and hence the probability of tail, q =1/2

For exactly two heads:

P(x=2) = 5 C2 p 2 q 5-2 = 5! / 2! 3! × (½) 2 × (½) 3

P(x=2) = 5/16

(b) For at least four heads,

x ≥ 4, P(x ≥ 4) = P(x = 4) + P(x=5)

P(x = 4) = 5 C4 p 4 q 5-4 = 5!/4! 1! × (½) 4 × (½) 1 = 5/32

P(x = 5) = 5 C5 p 5 q 5-5 = (½) 5 = 1/32

P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16

Example 2: For the same question given above, find the probability of:

a) Getting at most 2 heads

Solution: P (at most 2 heads) = P(X ≤ 2) = P (X = 0) + P (X = 1) + + P (X = 2)

P(X = 0) = (½) 5 = 1/32

P(X=1) = 5 C 1 (½) 5. = 5/32

P(x=2) = 5 C2 p 2 q 5-2 = 5! / 2! 3! × (½) 2 × (½) 3 = 5/16

P(X ≤ 2) = 1/32 + 5/32 + 5/16 = 1/2

A fair coin is tossed 10 times, what are the probability of getting exactly 6 heads and at least six heads.

Let x denote the number of heads in an experiment.

Here, the number of times the coin tossed is 10. Hence, n=10.

The probability of getting head, p ½

The probability of getting a tail, q = 1-p = 1-(½) = ½.

The binomial distribution is given by the formula:

P(X= x) = n C x p x q n-x , where = 0, 1, 2, 3, …

Therefore, P(X = x) = 10 C x (½) x (½) 10-x

(i) The probability of getting exactly 6 heads is:

P(X=6) = 10 C 6 (½) 6 (½) 10-6

P(X= 6) = 10 C 6 (½) 10

P(X = 6) = 105/512.

Hence, the probability of getting exactly 6 heads is 105/512.

(ii) The probability of getting at least 6 heads is P(X ≥ 6)

P(X ≥ 6) = P(X=6) + P(X=7) + P(X= 8) + P(X = 9) + P(X=10)

P(X ≥ 6) = 10 C 6 (½) 10 + 10 C 7 (½) 10  + 10 C 8 (½) 10  + 10 C 9 (½) 10  + 10 C 10 (½) 10

P(X ≥ 6) = 193/512.

Solve the following problems based on binomial distribution:

• The mean and variance of the binomial variate X are 8 and 4 respectively. Find P(X<3).
• The binomial variate X lies within the range {0, 1, 2, 3, 4, 5, 6}, provided that P(X=2) = 4P(x=4). Find the parameter “p” of the binomial variate X.
• In binomial distribution, X is a binomial variate with n= 100, p= ⅓, and P(x=r) is maximum. Find the value of r.

Frequently Asked Questions on Binomial Distribution

What is meant by binomial distribution.

The binomial distribution is the discrete probability distribution that gives only two possible results in an experiment, either success or failure.

Mention the formula for the binomial distribution.

The formula for binomial distribution is: P(x: n,p) = n C x p x (q) n-x Where p is the probability of success, q is the probability of failure, n= number of trials

What is the formula for the mean and variance of the binomial distribution?

The mean and variance of the binomial distribution are: Mean = np Variance = npq

What are the criteria for the binomial distribution?

The number of trials should be fixed. Each trial should be independent. The probability of success is exactly the same from one trial to the other trial.

What is the difference between a binomial distribution and normal distribution?

The binomial distribution is discrete, whereas the normal distribution is continuous.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

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Binomial Distribution

Introduction, binomial distribution definition , binomial distribution function, characteristics of the binomial distribution, negative binomial distribution , binomial distribution example , limitations and assumptions, alternatives for non-binomial scenarios, wrapping up .

Binomial distribution models the number of successes in a fixed number of independent trials with constant probability. The binomial distribution's roots trace back to the work of Swiss mathematician Jacob Bernoulli in the late 17th century. The concept has undergone subsequent improvements by Laplace, Pascal, and others.

The binomial distribution is fundamental in probability theory and statistics today. It is the cornerstone for modeling discrete random variables and precisely analyzing real-world phenomena. Here is a guide to the binomial distribution formula and its applications.

The independent Bernoulli trials each contain the same probability of success. The section below covers the basic concepts of binomial distribution. 

Basic Concepts of Binomial Distribution 

1. Bernoulli Trials

Bernoulli trials are experiments or processes that have two possible outcomes. The outcomes of each trial can result in success or failure. Each trial is independent, and the result of one trial does not affect subsequent trials. A good example is flipping a coin (heads or tails) or rolling a die (success if a specific number comes up).

2. Trial Outcomes (Success and Failure)

Success typically represents the occurrence of an event of interest, while failure represents the event's non-occurrence. These outcomes are mutually exclusive, and only one of them can occur in a single trial.

3. Independence of Trials

Independence of trials means that the outcome of one trial does not influence the result of any other trial. Each trial is subject to the same conditions and probability of success or failure.

4. Probability of Success (p) and Failure (q)

You can use  'p ' to denote the probability of success and 'q ' to denote the probability of failure. The probabilities of success and failure must sum to ( 1: p + q = 1 ). The probability 'p ' represents the likelihood of the desired outcome (success), while 'q ' represents the likelihood of the alternative outcome (failure).

The binomial distribution function is responsible for calculating the probability of obtaining a specific number of successes in a fixed number of independent Bernoulli trials with a constant success probability.

Binomial Distribution Formula

Below is a mathematical representation of the probability mass function (PMF). 

P(X-k) - (nk) * Pk * (1-p)n-k

• P(X-k) is the probability of getting exactly 𝑘 successes in 𝑛 trials.
• (nk) is the binomial coefficient. It represents the number of ways to choose 𝑘 successes out of 𝑛 trials.
• (p) is the probability of success in a single trial.
• (1 - p) is the probability of failure in a single trial.
• k ranges from 0 to 𝑛 inclusive.

Binomial Distribution Probability Formula

The binomial distribution formula involves two main parameters and below is a summary. 

• n (number of trials) : This parameter represents the total number of independent trials or experiments.
• p (probability of success) : This parameter represents the probability of success for an individual trial. It is the probability of the event of interest occurring in a single trial.

Cumulative Distribution Function (CDF)

Below is the mathematical representation of the cumulative distribution function. 

Cumulative distribution function (CDF)

F(k)=P(X≤k)=∑i=0k​(in​)×pi×(1−p)n−i

• F( k ) is the cumulative probability up to 𝑘 .
• P(X ≤ k ) represents the probability that the number of successes is less than or equal to 𝑘 .
• (ni) is the binomial coefficient. It represents the number of ways to choose 𝑖 successes out of 𝑛 trials.
• k ranges from 0 to 𝑛 , inclusive.

The CDF provides a way to assess the probability of achieving a certain number of successes or fewer in a binomial experiment. It is an essential method for analyzing the distribution of outcomes over a range of values.

The characteristics encompass fundamental properties, including probability distribution properties, mean and variance, skewness and kurtosis, and mode. Here is a highlight of the binomial distribution properties. 

Probability Distribution Properties

The binomial distribution exhibits several key probability distribution properties.

• Discreteness: The binomial distribution is a discrete probability distribution because it describes the probabilities of discrete outcomes (such as the number of successes in a fixed number of trials).
• Finite Support: The support of the binomial distribution is finite, ranging from 0 to 𝑛 inclusive, where 𝑛 is the number of trials in the experiment. 
• Non-Negative Probabilities: The probabilities assigned by the binomial distribution are non-negative, ensuring that the probability of observing any particular outcome is always greater than or equal to zero.
• Sum of Probabilities: The sum of probabilities for all possible outcomes equals 1.

Mean and Variance

The mean (μ) and variance (2) are important characteristics that describe the central tendency and spread of the distribution.

• Mean: You can calculate it as the product of the number of trials ( 𝑛) and the probability of success (𝑝): μ=n×p . The mean represents the average number of successes expected in 𝑛 trials.
• Variance ( σ 2 ): You can calculate it as the product of the number of trials ( 𝑛 ), the probability of success ( 𝑝 ), and the probability of failure: (1−𝑝): 𝜎2= 𝑛 × 𝑝 × (1−𝑝). The variance measures the spread or dispersion of the distribution around its mean. A larger variance indicates greater variability in the number of successes observed across trials.

Skewness and kurtosis

Skewness and kurtosis are two important characteristics that describe the shape of the distribution.

• Skewness: Measures the asymmetry of the probability distribution. Below is a mathematical representation of skewness in a binomial distribution. The magnitude of skewness decreases as 𝑛 increases, leading to a more symmetric distribution for large 𝑛.

• Kurtosis: Kurtosis measures the "tailedness" or the peakedness of the probability distribution. Below is a mathematical representation of kurtosis and excess kurtosis (kurtosis relative to a normal distribution). A binomial distribution with low 𝑛 can exhibit higher kurtosis, indicating a more peaked distribution with heavier tails. As 𝑛 increases, the kurtosis approaches that of a normal distribution, particularly when 𝑝 is not too close to 0 or 1.

The mode is the value(s) of the random variable 𝑋 that has the highest probability of occurring. You can calculate the mode by using the following mathematical equation. The ⌊x⌋ denotes the floor function, which rounds down 𝑥 to the nearest integer.

Mode = [(n + 1)p] or [(n+1)p] -1

If (𝑛+1) is an integer, then the binomial distribution has two modes: (𝑛+1)𝑝 and ( n +1) p −1. If (𝑛+1) is not an integer, then the binomial distribution has a single mode a ⌊(𝑛+1)𝑝⌋.

A negative binomial distribution is a discrete probability distribution that models the number of trials needed to achieve a specified number of successes in a sequence of independent and identically distributed Bernoulli trials. The mathematical formula below illustrates the probability mass function of a negative binomial distribution. 

• (k + r - 1k) is the binomial coefficient, representing the number of ways to arrange 𝑘 failures and 𝑟 successes in 𝑘+𝑟 trials.
• k is the number of failures, and 𝑋 represents the random variable for the number of failures before the 𝑟 -th success.

Real-world applications for the binomial distribution are wide. Below is a summary of areas of application. 

• Coin Toss: Calculating the probability of getting a certain number of heads in multiple coin tosses.
• Quality Control: Determining the probability of finding a certain number of defective items in a batch.
• Survey Results: Estimating the number of people who will respond positively in a sample survey given a probability of a positive response.
• Clinical Trials: Evaluating the success rate of a new drug by comparing the number of patients who improve (success) to the number who do not (failure).
• Market Research: Determining the likelihood that a certain number of customers will prefer a new product over an existing one in a given sample size.
• Genetics: Calculating the probability of inheriting a specific trait based on Mendelian genetics.
• Epidemiology: Estimating the likelihood of a certain number of individuals contracting a disease in a population given an infection probability.
• Stock Market: Predicting the number of days a stock will close above a certain price in a month.
• Risk Management: Assessing the probability of defaults in a portfolio of loans.

The binomial distribution has drawbacks because it relies on certain assumptions to work. The assumptions include a fixed number of trials (𝑛), two possible outcomes, constant probability of success (p), and independence of trials. The assumptions make the method unreliable in certain situations, as illustrated below.

• Variable Probability of Success: The binomial distribution is not suitable if the probability of success changes from trial to trial. 
• More Than Two Outcomes: The binomial distribution is inappropriate if each trial can result in more than two outcomes (e.g., dice rolling). 
• Dependent Trials: The binomial distribution does not apply if the trials are not independent. 

Here are a couple of alternatives for non-binomial scenarios. 

Hypergeometric Distribution: Efficient method for sampling without replacement from a finite population.

Negative Binomial Distribution: You can use it to count the number of trials needed to achieve a fixed number of successes.

Multinomial Distribution: Generalizes the binomial distribution for scenarios with more than two possible outcomes for each trial.

Normal Distribution: Efficient when the number of trials is large, and both 𝑛𝑝 and (1−𝑝) are greater than 5, the normal distribution can approximate the binomial distribution (Central Limit Theorem).

The binomial distribution has proven to be a crucial statistical tool for modeling or analyzing the probability of a fixed number of successes in a series of independent trials with constant success probability. You can see the use of binomial distribution in financial institutions, biology research centers, and quality control sectors. 

1. What are the 4 properties of the binomial distribution?

Four properties of binomial distribution include a fixed number of trials, two possible outcomes, constant probability of success, and independence of trials. 

2. What is the full formula of binomial distribution?

The formula of the binomial distribution is P(X - k) - (nk) Pk(1-P)n-k.

3. What are the main features of binomial distribution?

The main features of the binomial distribution are discrete probability distribution, a fixed number of trials, two outcomes, and a constant probability of success. 

4. What are the types of binomial distribution?

The types of binomial distribution are symmetric, positively skewed, and negatively skewed. 

5. What is a binomial distribution with an example?

A binomial distribution models the number of successes in a fixed number of independent trials, like flipping a coin.

6. What is the use of binomial distribution?

The binomial distribution is an efficient method for predicting the number of successful outcomes in repeated trials with fixed probability.

7. What is the real-life application of binomial distribution?

The binomial distribution helps in fields like quality control to predict the number of defective items in a batch.

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Column: What kind of prosecutor was Kamala Harris? The answer could be pivotal to her campaign

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The presidential campaign is shaping up to be a tale of two prosecutors, both named Kamala Harris.

Harris’ record as a tough-on-crime prosecutor was a constant refrain through much of this week’s Democratic National Convention.

On Monday, the convention featured an ad parodying the television series “Law & Order” that asserted, “We need a president who has spent her life prosecuting perpetrators like Donald Trump.” Tuesday’s program was even more blunt, including a five-minute video showcasing Harris as a crusading prosecutor who unflinchingly locked up murderers, rapists and child molesters.

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A series of speakers echoed that theme, touting Harris’ lock-’em-up bona fides. In his headlining speech Tuesday, former President Obama cast Harris, a former San Francisco district attorney and California attorney general, as an uncompromising prosecutor. As with the string of speakers making similar points, Obama focused on red-meat invocations of rapists, child molesters and other sexual predators.

Harris’ experience as a law woman has been the leading theme of her young campaign since she made her debut and rolled out the reliable applause line, “I took on perpetrators of all kinds. … So hear me when I say: I know Donald Trump’s type.”

It all sounds like a vintage campaign for district attorney — and the kind of rhetoric much more frequently deployed by Republicans.

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Yet even as Harris is selling her prosecutorial credentials, the Trump campaign is trotting out a vision of her as feckless and soft on crime — indeed, “one of the worst prosecutors in history,” according to Trump’s characteristically fact-free hyperbole.

This struggle over Harris’ prosecutorial identity is for now the core of the campaign, an unusual feature for a presidential contest. But given Trump’s complete familiarity to supporters and opponents alike, the main variable in the campaign may well be whose vision of Harris prevails.

Trump’s caricature of Harris’ record relies on a series of deceptive and false claims. He has asserted with no basis whatsoever that as district attorney, she “ wouldn’t arrest murderers . She wouldn’t arrest anybody.”

Trump also tells his supporters that Harris “supports mandatory gun confiscation” that would leave Americans “defenseless.” That refers to a mandatory assault weapons buyback program that Harris once supported but no longer does.

On immigration law, Trump’s line is that Harris wanted to give “mass amnesty and citizenship” to “all illegals.” The basis for this claim is her past support for a path to citizenship and amnesty for limited groups such as the so-called Dreamers brought into the country illegally as children.

Or Trump accuses Harris of redefining “child sex trafficking, assault with a deadly weapon and rape of a unconscious person … as nonviolent,” a whopper that apparently refers to a 2016 California voter referendum that enabled early release consideration for those convicted of “nonviolent felonies.”

Trump also argues that Harris “supports abolishing cash bail,” which is in fact true. But his conclusion — “which means bloodthirsty criminals that just killed somebody can immediately leave custody, go out and kill somebody else” — is false. Rather, Harris and others support replacing traditional bail with other standards for detaining violent offenders before trial.

You get the idea: Combine the emotional politics of violent crime with Trump’s indifference to truth, and you get a volatile brew of sensationalist accusations that make the George H.W. Bush campaign’s infamous Willie Horton attack look like beanbag. But as the Horton episode demonstrated, deceptive claims about crime can be sticky.

And Harris has a complicated track record on the subject. When she first ran for president, in 2020, she adopted the identity of a “progressive prosecutor,” more or less the opposite of the portrait the convention painted this week. That drew fire from the left based on her past support of initiatives that might strike progressive voters as overly punitive, especially in the wake of that year’s police killing of George Floyd. She also encountered opposition from police groups based on her long-standing opposition to the death penalty.

Four years later, having suddenly acquired the nomination in the wake of President Biden’s disastrous debate, Harris is no longer shying away from the characterizations that undermined her candidacy with the left as she competed in the crowded 2020 primary.

In fact, Harris’ enthusiastic donning of the mantle of tough prosecutor is of a piece with a broader theme of the convention that likely will carry on into the campaign: the Democrats’ self-conscious effort to reclaim an array of traditional American virtues. The first half of the convention was marked by appeals to a throwback, “ Ozzie and Harriet ” world of Little League games and church socials. Obama put it best in his patent appeal to swing or even Trump voters, saying the majority of the American people “do not want to live in a country that’s bitter and divided.”

Harris is embracing that communitarian vision even as she is presenting herself as a zealous prosecutor of those who would disrupt it. Whether she can maintain that image in the face of Trump’s darker vision may be the central drama of the campaign ahead.

Harry Litman is the host of the “Talking Feds” podcast and the “ Talking San Diego” speaker series. @harrylitman

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Harry Litman, the senior legal affairs columnist for the Opinion page, is a former U.S. attorney and deputy assistant attorney general. He is the creator and host of the “Talking Feds” podcast ( @talkingfedspod ). Litman teaches constitutional and national security law at UCLA and UC San Diego and is a regular commentator on MSNBC, CNN and CBS News.

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