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Statistics By Jim

Making statistics intuitive

Binomial Distribution Formula: Probability, Standard Deviation & Mean

By Jim Frost 2 Comments

Binomial Distribution Formula

Use the binomial distribution formula to calculate the likelihood an event will occur a specific number of times in a set number of opportunities. I’ll show you the binomial distribution formula to calculate these probabilities manually.

In this post, I’ll walk you through the formulas for how to find the probability, mean, and standard deviation of the binomial distribution and provide worked examples.

Note that this post focuses on the binomial distribution formulas and calculations. For more information about the distribution itself and how to use and graph it, please read Binomial Distribution: Uses & Calculator .

The binomial distribution formula for probabilities is the following:

• n is the number of trials.
• x is the number of successes
• p is the probability of a success.
• (1–p) is the chance of failure.

Use this formula to calculate the binomial probability for X successes occurring in n trials.

nCx is the number of ways to obtain samples with the specified number of successes occurring within the set number of trials where the order of outcomes does not matter. Specifically, it’s the number of combinations without repetition. For more information, read my post about Finding Combinations .

The binomial distribution formula takes the number of combinations, multiplies that by the probability of success raised by the number of successes, and multiplies that by the probability of failures raised by the number of failures.

Let’s work through an example calculation to bring the formula to life!

Worked Example of Finding a Binomial Probability

We’ll use the binomial distribution formula to calculate the chances of rolling exactly three sixes in ten die rolls for this example. Here are the values to enter into the formula:

• (1–p) = 0.8333

For the number of combinations, we have:

Now, let’s enter our values into the binomial distribution formula.

This calculation finds that the binomial probability of rolling three 6s in 10 rolls is 0.1540.

If you need to calculate a cumulative probability for a binomial random variable, calculate the likelihood for each individual outcome and then sum them for all outcomes of interest.

For example, if you want to calculate the probability of ≥ 3 sixes in 10 rolls, calculate the likelihoods for three sixes, four sixes, etc., on up to ten sixes. Then sum that set of binomial probabilities.

Read on to learn about the formulas to calculate the mean and standard deviation of the binomial distribution!

Expected Value of Binomial Distribution

The expected value of the binomial distribution is its mean.

The binomial distribution formula for the expected value is the following:

Multiply the number of trials (n) by the success probability (p). This value represents the average or expected number of successes.

For example, we roll the die ten times, and the probability of rolling a six is 0.1667. Let’s enter these values into the formula

10 * 0.1667

The mean for this binomial distribution is 1.667. On average, we’d expect to roll that many sixes in ten rolls. Of course, the actual counts of successes will always be either zero or a positive integer.

This mean is the expected value for a binomial distribution. Learn more about Expected Values: Definition, Formula & Finding .

Standard Deviation of the Binomial Distribution

The binomial distribution formula for the standard deviation is the following:

As before, n and p are the number of trials and success probability, respectively. (1 – p) is the likelihood of failure.

Notice that the standard deviation of the binomial distribution is at its maximum when the probabilities for success and failure are both 0.5. As those probabilities move away from 0.5 in opposite directions, it decreases. Additionally, it increases as the number of trials (n) increase.

For our die example we have n = 10 rolls, a success probability of p = 0.1667, and a failure probability of (1 – p) = 0.833. Let’s enter these values into the formula.

10 * 0.1667 * 0.8333 = 1.3891. That’s the variance , which uses squared units.

To find the standard deviation of the binomial distribution, we need to take the square root of the variance.

The standard deviation represents the variability of the probabilities around the mean of the binomial distribution. Learn more about the Standard Deviation .

By using the formula for the binomial distribution, it is easy to calculate its probabilities, means, and standard deviations.

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Reader Interactions

June 13, 2024 at 2:31 am

I would like to have a question regarding the distribution of outcomes of multiple binomial distributions.

I have a sample of 50 subjects, where every subject completes a task with two possible outcomes (left or right hand use, with 50% probability) 30 times. On an individual level, this leads to a binomial distribution of outcomes for each subject, where a z-score can be calculated for each subject to see how far the hand uses are from the expected 50% (15 left and right hand use) level. Based on the z-scores, we deem those with larger values than 1.96 (›1.96) as right-handed, those with lower values than -1.96 (‹-1.96) as left-handed and those with values in between (-1.96‹‹1.96) as ambilateral.

Question: what is the expected ratio of z-scores (or distribution of z-scores) in my sample of 50 subjects? Is it the same as at the individual level with 95% chance of being ambilateral and 2.5-2.5% as being left or right-handed?

December 14, 2023 at 5:30 pm

I have a probability question I have been trying to figure out. Here is the scenario: A group (12) of golfers gather each Saturday to play in a match. Each player is trying to gain points based on their individual handicap. For example, my handicap is 34 and if I get 34 pts I finish Even for the day. If I get 32 pts, I finish -2; 36 pts, +2 and so on.

At the end of the match players are randomly drawn as teammates from a hat. The team with the highest net score wins the match. Let’s assume for this match, teams will be drawn out in 2s. Let’s also assume that the maximum + or – for each player is 5. For instance, I may be drawn with player #7 who is +4 for the day and I am -1 for the day. Our team score is +3.

What is the probability that I (my team) will win? Do my odds change based on my score?

What if there are 20 players? Or, if teams are divided into 4s?

Thanks for providing some help and insight!

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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
• Poisson Distribution

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Chapter 4: Discrete Random Variables

4.3 Binomial Distribution

Learning objectives.

By the end of this section, you should be able to:

• Identify the components of a binomial experiment
• Use the formulas for a binomial random variable to compute mean, variance, and standard deviation

Binomial Experiments

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 Taylor consistently guesses correctly on any statistics true-false question with probability [latex]p = 0.6[/latex] . Then, [latex]q = 0.4.[/latex] Consistency in guessing means that for every true-false statistics question Taylor answers, the probability of success ([latex]p = 0.6[/latex] ) and the probability of failure ([latex]q = 0.4[/latex] ) remain the same, a necessary criteria for a situation to be binomial.

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.

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.

Notation for the Binomial

We use [latex]B[/latex] to represent the binomial probability distribution, and when [latex]X[/latex] fits the binomial distribution, we write [latex]X \sim B(n,p)[/latex]. Read this as “[latex]X[/latex] is a random variable with a binomial distribution with parameters [latex]n[/latex] and [latex]p[/latex],”  which again represent the number of trials and the probability of “success.”

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.

If at the start of a particular term, 300 students are enrolled in the elementary physics course, [latex]n = 300[/latex] and [latex]p = 0.3[/latex] and [latex]X \sim B(300, 0.3)[/latex]. The possible outcomes are [latex]x = 0, \ldots, 300[/latex] and the probability [latex]P(X=x)[/latex] is the probability that [latex]x[/latex] students will withdraw during the term.

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?

a. This is a binomial problem because there is only a success or a [latex]\underline{\hspace{20pt}}[/latex], 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 [latex]X[/latex]?

c. What values does [latex]x[/latex] take on?

d. What is a “failure,” in words?

e. If [latex]p + q = 1[/latex], then what is [latex]q[/latex]?

f. The words “at least” translate as what kind of inequality for the probability question [latex]P(X \underline{\hspace{20pt}} 40)[/latex].

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.

Experiments That Are Not Binomial

Here are some common experiments that are not binomial:

• Flipping a coin until you get one head. While there are only two outcomes (heads and tails) to each coin flip, and while the probability of getting a head on each flip is consistent, the number of trials will vary.
• Most characteristics about people, such as weight, height, ethnicity, and gender. When we survey a group of people and ask for their weight, we get far more than two responses. When surveying ethnicity, as discussed in Chapter 1 an Other or Unknown category is needed to include all people in our results, specifically people who did not feel they fit into any of the ethnicity categories or declined to respond. While gender has historically been treated as a binomial outcome (male and female), not all people will feel they fit those two categories, so we may lose information and reliability by modeling gender as binomial.
• Drawing cards without replacement. While you can specify a set number of [latex]n[/latex] trials, and you can set it up as two outcomes (success = red, failure = black), the draws are not independent. If you draw a red on the first draw, it is now slightly more likely to draw black than red on the second draw.

ABC College has a student advisory committee made up of ten staff members and six students. The committee wishes to choose a chairperson and a recorder by putting the names of all committee members into a box, and two names are drawn without replacement. The first name drawn determines the chairperson and the second name the recorder. Suppose the college wishes to find the probability that the chairperson and recorder are both students. Is this binomial?

Suppose 55% of people pass the state driver’s exam on the first try. You survey fifty randomly chosen adults in the state.

Which of these are binomial problems?

• The number of the adults who have a driver’s license.
• The number of the adults who got their driver’s license on the first try.
• The number of times each adult has taken the driver’s exam.

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, [latex]P(X=x)[/latex].  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:

nCx = [latex]=\binom nx =\frac{n!}{x!(n-x)!}[/latex]

Note: 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 ([latex]p[/latex]) raised to the number of successes ([latex]x[/latex]) by the probability of failure ([latex]q=1-p[/latex]) raised to the number of failures ([latex]n-x[/latex]).

[latex]p^x q^{(n-x)}[/latex]

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.

[latex]P(X = x) = \frac{n!}{x!(n-x)!} p^x q^{(n-x)}[/latex]

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. Twenty adult workers are randomly selected.

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

Then [latex]X[/latex] takes on the values [latex]0, 1, 2, \ldots, 20[/latex] where [latex]n = 20, p = 0.41[/latex], and [latex]q = 1-0.41 = 0.59[/latex]. Finally, [latex]X \sim B(20,0.41).[/latex]

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

The graph of [latex]X \sim B(20, 0.41)[/latex] 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?

To compute binomial probabilities on a graphing calculator, g o into 2 nd DISTR. The syntax for the instructions are as follows:

To calculate [latex]P(X = x)[/latex]: binompdf( n , p , x). If [latex]x[/latex] is left out, the result is the binomial probability table.

To calculate [latex]P(X \leq x)[/latex]: binomcdf( n , p , x). If [latex]x[/latex] is left out, the result is the cumulative binomial probability table.

For the example problem: After you are in 2 nd DISTR, arrow down to binomcdf(. Press ENTER. Enter 20,0.41,12). The result is [latex]P(X \leq 12) = 0.9738[/latex].

If you wanted to instead find [latex]P(X>12)[/latex], use 1 – binomcdf(20,0.41,12).

In Excel, both binomial probabilities are computed using BINOM.DIST(x, n, p, True/False), where False computes [latex]P(X=x)[/latex] and is equivalent to binompdf, and True computes [latex]P(X \leq x)[/latex] and is equivalent to binomcdf.

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.

Expected Value and Standard Deviation

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 is then [latex]\sigma = \sqrt{npq}[/latex].

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?
• the probability that two pages feature signature artists
• the probability that at most six pages feature signature artists
• the probability that more than three pages feature signature artists.
• Using the formulas, calculate the (i) mean and (ii) standard deviation.
• x = 0, 1, 2, 3, 4, 5, 6, 7, 8
• P ( x = 2) = binompdf[latex]\left(100,\frac{8}{560},2\right)[/latex] = 0.2466
• P ( x ≤ 6) = binomcdf[latex]\left(100,\frac{8}{560},6\right)[/latex] = 0.9994
• P ( x > 3) = 1 – P ( x ≤ 3) = 1 – binomcdf[latex]\left(100,\frac{8}{560},3\right)[/latex] = 1 – 0.9443 = 0.0557
• Mean = np = (100)[latex]\left(\frac{8}{560}\right)[/latex] = [latex]\frac{800}{560}[/latex] ≈ 1.4286
• Standard Deviation = [latex]\sqrt{npq}[/latex] = [latex]\sqrt{\left(100\right)\left(\frac{8}{560}\right)\left(\frac{552}{560}\right)}[/latex] ≈ 1.1867

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.

During the 2013 regular NBA season, DeAndre Jordan of the Los Angeles Clippers had the highest field goal completion rate in the league. DeAndre scored with 61.3% of his shots. Suppose you choose a random sample of 80 shots made by DeAndre during the 2013 season. Let X = the number of shots that scored points.

• Using the formulas, calculate the (i) mean and (ii) standard deviation of X .
• Use your calculator to find the probability that DeAndre scored with 60 of these shots.
• Find the probability that DeAndre scored with more than 50 of these shots.

“Access to electricity (% of population),” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/EG.ELC.ACCS.ZS?order=wbapi_data_value_2009%20wbapi_data_value%20wbapi_data_value-first&sort=asc (accessed May 15, 2015).

“Distance Education.” Wikipedia. Available online at http://en.wikipedia.org/wiki/Distance_education (accessed May 15, 2013).

“NBA Statistics – 2013,” ESPN NBA, 2013. Available online at http://espn.go.com/nba/statistics/_/seasontype/2 (accessed May 15, 2013).

Newport, Frank. “Americans Still Enjoy Saving Rather than Spending: Few demographic differences seen in these views other than by income,” GALLUP® Economy, 2013. Available online at http://www.gallup.com/poll/162368/americans-enjoy-saving-rather-spending.aspx (accessed May 15, 2013).

Pryor, John H., Linda DeAngelo, Laura Palucki Blake, Sylvia Hurtado, Serge Tran. The American Freshman: National Norms Fall 2011 . Los Angeles: Cooperative Institutional Research Program at the Higher Education Research Institute at UCLA, 2011. Also available online at http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf (accessed May 15, 2013).

“The World FactBook,” Central Intelligence Agency. Available online at https://www.cia.gov/library/publications/the-world-factbook/geos/af.html (accessed May 15, 2013).

“What are the key statistics about pancreatic cancer?” American Cancer Society, 2013. Available online at http://www.cancer.org/cancer/pancreaticcancer/detailedguide/pancreatic-cancer-key-statistics (accessed May 15, 2013).

Media Attributions

an experiment with the following characteristics: 1. There are only two possible outcomes called “success” and “failure” for each trial. 2. 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 statistical experiment that satisfies three conditions: 1. There are a fixed number of trials, n . 2. 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. 3. The n trials are independent and are repeated using identical conditions.

A discrete random variable which arises from Bernoulli trials; there are a fixed number, n , of independent trials with two outcomes called success and failure with probability p and q respectively. The binomial random variable X is the number of successes in n trials, denoted [latex]X \sim B(n,p)[/latex]. The mean is [latex]\mu = np[\latex] and the standard deviation is [latex]\sigma = \sqrt{npq}[/latex]. The probability of exactly [latex]x[/latex] successes in n trials is [latex]P\left(X=x\right)=\left(\genfrac{}{}{0}{}{n}{x}\right){p}^{x}{q}^{n-x}[/latex].

A function that gives the probability that a discrete random variable (X) 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

Introductory Statistics Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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Binomial Distribution

These lessons, with videos, examples and step-by-step solutions, help Statistics students learn how to use the binomial distribution.

Related Pages Binomial Distribution Tutorial Binomial Distribution: Critical Values More Lessons for Statistics Math Worksheets

Perhaps the most widely known of all discrete distribution is the binomial distribution. The binomial distribution has been used for hundreds of years. Several assumptions underlie the use of the binomial distribution.

Assumptions of the binomial distribution:

• The experiment involves n identical trials.
• Each trial has only two possible outcomes denoted as success or failure.
• Each trial is independent of the previous trials.
• The terms p and q remain constant throughout the experiment, where p is the probability of getting a success on any one trial and q = (1 – p) is the probability of getting a failure on any one trial.

The following diagram gives the Binomial Distribution Formula. Scroll down the page for more examples and solutions.

Introduction to the binomial distribution

More Binomial Distribution

Basketball binomial distribution

Using Excel to visualize the basketball binomial distribution

Binomial Experiment The following video will discuss what a binomial experiment is, discuss the formula for finding the probability associated with a binomial experiment, and gives an example to illustrate the concepts.

Example: Suppose you take a multiple choice test with 10 questions, and each question has 5 answer choices (a, b, c, d, e), what is the probability you get exactly 4 quaetions correct just by guessing?

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Binomial Distribution: Formula, What it is, How to use it

What is a binomial distribution, the binomial distribution formula.

• Worked Examples
• The Bernoulli Distribution

Lexian distribution

  The binomial distribution evaluates the probability for an outcome to either succeed or fail. These are called mutually exclusive outcomes, which means you either have one or the other — but not both at the same time. For example, you either win the lottery or you don’t, a drug to cure a disease works or it doesn’t, or a test results in a pass or a fail. Binomial distributions come help us to analyze the probability of events such as these. Many instances of binomial distributions can be found in real life. For example, if a new drug is introduced to cure a disease, it either cures the disease (it’s successful) or it doesn’t cure the disease (it’s a failure). If you purchase a lottery ticket, you’re either going to win money, or you aren’t. Basically, anything you can think of that can only be a success or a failure can be represented by a binomial distribution.

Binomial distributions must meet the following three criteria:

• The number of observations or trials is fixed. In other words, you can only figure out the probability of something happening if you do it a certain number of times. This is common sense—if you toss a coin once, your probability of getting a tails is 50%. If you toss a coin a 20 times, your probability of getting a tails is very, very close to 100%.
• Each observation or trial is independent . In other words, none of your trials have an effect on the probability of the next trial.
• The probability of success (tails, heads, fail or pass) is exactly the same from one trial to another.

Once you know that your distribution is binomial, you can apply the binomial distribution formula to calculate the probability.

The binomial distribution formula is: b(x; n, P) = n C x * P x * (1 – P) n – x Where:

• b = binomial probability
• n C x = combinations formula n C x = n! / (x!(n – x)!)
• x = total number of “successes”
• P = probability of a success on a single attempt
• n = number of attempts or trials.

Binomial Distribution Formula Examples

The binomial distribution formula can calculate the probability of success for binomial distributions. Often you’ll be told to “plug in” the numbers to the formula and calculate . This is easy to say, but not so easy to do—unless you are very careful with order of operations , you won’t get the right answer. If you have a Ti-83 or Ti-89, the calculator can do much of the work for you. If not, here’s how to break down the problem into simple steps so you get the answer right—every time.

Q. A coin is tossed 10 times. What is the probability of getting exactly 6 heads? I’m going to use this formula: b(x; n, P) – n C x * P x * (1 – P) n – x The number of trials (n) is 10 The odds of success (“tossing a heads”) is 0.5 (So 1-p = 0.5) x = 6 P(x=6) = 10 C 6 * 0.5 6 * 0.5 4 = 210 * 0.015625 * 0.0625 = 0.205078125 Tip: You can use the combinations calculator to figure out the value for n C x .

Q. 80% of people who purchase pet insurance are women.  If 9 pet insurance owners are randomly selected, find the probability that exactly 6 are women.

• Identify ‘n’ from the problem. Using our example question, n (the number of randomly selected items) is 9.
• Identify ‘X’ from the problem. X (the number you are asked to find the probability for) is 6.
• Work the first part of the formula. The first part of the formula is n! / (n – X)!  X! Substitute your variables: 9! / ((9 – 6)! × 6!) Which equals 84. Set this number aside for a moment.
• Find p and q. p is the probability of success and q is the probability of failure. We are given p = 80%, or .8. So the probability of failure is 1 – .8 = .2 (20%).
• Work the second part of the formula. p X = .8 6 = .262144 Set this number aside for a moment.
• Work the third part of the formula. q (n – X) = .2 (9-6) = .2 3 = .008
• Multiply your answer from step 3, 5, and 6 together. 84  × .262144 × .008 = 0.176.

Q. 60% of people who purchase sports cars are men.  If 10 sports car owners are randomly selected, find the probability that exactly 7 are men.

• Identify ‘n’ and ‘X’ from the problem. Using our sample question, n (the number of randomly selected items—in this case, sports car owners are randomly selected) is 10,  and  X (the number you are asked to “find the probability” for) is 7.
• Figure out the first part of the formula, which is: n! / (n – X)!  X! Substituting the variables : 10! / ((10 – 7)! × 7!) Which equals 120. Set this number aside for a moment.
• Find “p” the probability of success and “q” the probability of failure. We are given p = 60%, or .6. therefore, the probability of failure is 1 – .6 = .4 (40%).
• Work the next part of the formula. p X = .6 7 = .0.0279936 Set this number aside while you work the third part of the formula.
• Work the third part of the formula. q (.4 – 7) = .4 (10-7) = .4 3 = .0.064
• Multiply the three answers from steps 2, 4 and 5 together. 120  × 0.0279936 × 0.064 = 0.215. That’s it!

The Bernoulli Distribution.

The binomial distribution is closely related to the Bernoulli distribution . According to Washington State University, “If each Bernoulli trial is independent, then the number of successes in Bernoulli trails has a binomial Distribution. On the other hand, the Bernoulli distribution is the Binomial distribution with n=1.”

A Bernoulli distribution is a set of Bernoulli trials. Each Bernoulli trial has one possible outcome, chosen from S, success, or F, failure. In each trial, the probability of success, P(S) = p, is the same. The probability of failure is just 1 minus the probability of success: P(F) = 1 – p. (Remember that “1” is the total probability of an event occurring — probability is always between zero and 1).

Finally, all Bernoulli trials are independent from each other and the probability of success doesn’t change from trial to trial, even if you have information about the other trials’ outcomes.

A Lexian distribution is another name for the binomial distribution ( k , p ) if p is not constant [1]. One way to interpret the distribution is as a special case of a mixture of binomial distributions [2], thus it is also called the mixed binomial distribution. The Lexian distribution considers a mixture distribution of subsets of binomials, each of which has its own probability distribution function (pdf).

The mean of the Lexian distribution is [3]

• n is the number of trials
• p̄ is the average value of the distinct probability distributions.

The Lexian variance is

• var( p ) is the variance of the average value of the distinct probability distributions.

As a consequence, if mixed binomial variables are treated as pure binomials, the mean would be correct but the variance would be underestimated when using the “binomial estimator” np (1- p ) [4].

History of the Lexian Distribution

The Lexian distribution is named after German economist Wilhelm Lexis , who published several papers on mixture distributions in 1875-1879. The basis of his work was to test for the structure of a set by comparing its actual variance to one obtained from a theoretical binomial variance through a “Lexis Ratio”: the standard deviation from the data, divided by the theoretical binomial standard deviation [5].

• ​wikipedia user Nusha, CC BY-SA 3.0 , via Wikimedia Commons
• Suchindran C. M. (1981). A reply to Avery and Hakkert. Population studies ,  35 (5), 473–475. https://doi.org/10.1080/00324728.1981.11878519
• Johnson, N. L. (1969), Discrete distributions, Houghton Mifflin Company, Boston.
• Coppens, F. et al. (2007). The performance of credit rating systems in the assessment of collateral used in Eurosystem monetary policy operations. National Bank of Belgium. Online: http://aei.pitt.edu/7612/1/wp118En.pdf
• Bensman, S. (2005). Urquhart’s Law: Probability and the Management of Scientific and Technical Journal Collections Part 1. The Law’s Initial Formulation and Statistical Bases. Haworth Press. doi:10.1300/J122v26n01_04

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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• 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!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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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% .

10 sided dice roller

Car crash force, quartic regression.

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What Is Binomial Distribution?

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Binomial Distribution: Definition, Formula, Analysis, and Example

Investopedia / Eliana Rodgers

Binomial distribution is a statistical distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters or assumptions.

The underlying assumptions of binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive or independent of one another.

Key Takeaways

• Binomial distribution is a statistical probability distribution that summarizes the likelihood that a value will take one of two independent values under a given set of parameters or assumptions.
• The underlying assumptions of binomial distribution are that there is only one outcome for each trial, that each trial has the same probability of success, and that each trial is mutually exclusive or independent of one another.
• Binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as normal distribution.

Understanding Binomial Distribution

To start, the “binomial” in binomial distribution means two terms—the number of successes and the number of attempts. Each is useless without the other.

Binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as normal distribution . This is because binomial distribution only counts two states, typically represented as 1 (for a success) or 0 (for a failure), given a number of trials in the data. Binomial distribution thus represents the probability for x successes in n trials, given a success probability p for each trial.

Binomial distribution summarizes the number of trials, or observations, when each trial has the same probability of attaining one particular value. Binomial distribution determines the probability of observing a specific number of successful outcomes in a specified number of trials.

Binomial distribution is often used in social science statistics as a building block for models for dichotomous outcome variables, such as whether a Republican or Democrat will win an upcoming election, whether an individual will die within a specified period of time, etc. It also has applications in finance, banking, and insurance, among other industries.

Analyzing Binomial Distribution

A binomial distribution's expected value, or mean, is calculated by multiplying the number of trials (n) by the probability of successes (p), or n × p.

For example, the expected value of the number of heads in 100 trials of heads or tails is 50, or (100 × 0.5). Another common example of binomial distribution is estimating the chances of success for a free-throw shooter in basketball, where 1 = a basket made and 0 = a miss.

The binomial distribution function is calculated as:

P ( x : n , p ) =  n C x p x ( 1 - p ) n - x

• n is the number of trials (occurrences)
• x is the number of successful trials
• p is the probability of success in a single trial
• n C x is the combination of n and x. A combination is the number of ways to choose a sample of x elements from a set of n distinct objects where order does not matter, and replacements are not allowed. Note that n C x = n! / r! ( n − r ) ! ), where ! is factorial (so, 4! = 4 × 3 × 2 × 1).

The mean of the binomial distribution is np, and the variance of the binomial distribution is np (1 − p). When p = 0.5, the distribution is symmetric around the mean—such as when flipping a coin because the chances of getting heads or tails is 50%, or 0.5. When p > 0.5, the distribution curve is skewed to the left. When p < 0.5, the distribution curve is skewed to the right.

The binomial distribution is the sum of a series of multiple independent and identically distributed Bernoulli trials. In a Bernoulli trial, the experiment is said to be random and can only have two possible outcomes: success or failure.

For instance, flipping a coin is considered to be a Bernoulli trial ; each trial can only take one of two values (heads or tails), each success has the same probability, and the results of one trial do not influence the results of another. Bernoulli distribution is a special case of binomial distribution where the number of trials n = 1.

Example of Binomial Distribution

Binomial distribution is calculated by multiplying the probability of success raised to the power of the number of successes and the probability of failure raised to the power of the difference between the number of successes and the number of trials. Then, multiply the product by the combination of the number of trials and successes.

For example, assume that a casino created a new game in which participants can place bets on the number of heads or tails in a specified number of coin flips. Assume a participant wants to place a $10 bet that there will be exactly six heads in 20 coin flips. The participant wants to calculate the probability of this occurring, and therefore, they use the calculation for binomial distribution.

The probability was calculated as (20! / (6! × (20 - 6)!)) × (0.50) (6) × (1 - 0.50) (20 - 6) . Consequently, the probability of exactly six heads occurring in 20 coin flips is 0.0369, or 3.7%. The expected value was 10 heads in this case, so the participant made a poor bet. The graph below shows that the mean is 10 (the expected value), and the chances of getting six heads is on the left tail in red. You can see that there is less of a chance of six heads occurring than seven, eight, nine, 10, 11, 12, or 13 heads.

StatCrunch Binomial Calculator

So how can this be used in finance? One example: Let’s say you’re a bank, a lender , that wants to know within three decimal places the likelihood of a particular borrower defaulting. What are the chances of so many borrowers defaulting that they would render the bank insolvent? Once you use the binomial distribution function to calculate that number, you have a better idea of how to price insurance and, ultimately, how much money to lend out and keep in reserve. 

Binomial distribution is a statistical probability distribution that states the likelihood that a value will take one of two independent values under a given set of parameters or assumptions.

How Is Binomial Distribution Used?

This distribution pattern is used in statistics but has implications in finance and other fields. Banks may use it to estimate the likelihood of a particular borrower defaulting, how much money to lend, and the amount to keep in reserve. It’s also used in the insurance industry to determine policy pricing and assess risk .

Why Is Binomial Distribution Important?

Binomial distribution is used to figure the probability of a pass or fail outcome in a survey, or experiment replicated numerous times. There are only two potential outcomes for this type of distribution. More broadly, distribution is an important part of analyzing data sets to estimate all the potential outcomes of the data and how frequently they occur. Forecasting and understanding the success or failure of outcomes is essential to business development.

The binomial distribution is an important statistical distribution that describes binary outcomes (such as the flip of a coin, a yes/no answer, or an on/off condition). Understanding its characteristics and functions is important for data analysis in various contexts that involve an outcome taking one of two independent values.

It has applications in social science, finance, banking, insurance, and other areas. For instance, it can be used to estimate whether a borrower will default on a loan, whether an options contract will finish in-the-money or out-of-the-money, or whether a company will miss or beat earnings estimates.

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Binomial Distribution Formula

The binomial distribution is a commonly used discrete distribution in statistics. The normal distribution as opposed to a binomial distribution is a continuous distribution. The binomial distribution represents the probability for 'x' successes of an experiment in 'n' trials, given a success probability 'p' for each trial at the experiment.

Binomial Distribution in Statistics:  The binomial distribution forms the base for the famous binomial test of statistical importance. A test that has a single outcome such as success/failure is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process. Consider an experiment where each time a question is asked for a yes/no with a series of n experiments. Then in the binomial probability distribution, the boolean-valued outcome the success/yes/true/one is represented with probability p and the failure/no/false/zero with probability q (q = 1 − p). In a single experiment when n = 1, the binomial distribution is called a Bernoulli distribution.

What Is the Binomial Distribution Formula?

The binomial distribution formula is for any random variable X, given by;  P(x:n,p) = n C\(_x\) p x (1-p) n-x Or P(x:n,p) = n C x p x (q) n-x

• 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 is also 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

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Examples on Binomial Distribution Formula

Example 1: If a coin is tossed 5 times, using binomial distribution 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

Answer: Therefore, P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16

Example 2: For the same question given above, find the probability of getting at most 2 heads.

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

P(X = 0) = (½) 5 = 1/32

P(X=1) = 5C1 (½) 5 .= 5/32

Answer: Therefore, P(X ≤ 2) = 1/32 + 5/32 = 3/16

Example 3: 60% of people who purchase sports cars are men. Find the probability that exactly 7 are men if 10 sports car owners are randomly selected.

Solution: Let’s Identify ‘n’ and ‘X’ from the problem. The number of sports car owners are randomly selected is n = 10,  and The number to find the probability is X = 7.

Given: p = 60%, or 0.6.  Therefore, the probability of failure is q = 1 – 0.6 = 0.4  Now, using the binomial distribution formula \( P( x ) = \frac{{n!}}{{( {n - x} )!x!}}.( p )^x .( q)^{n - x}  \\    = \frac{{10!}}{{( {10 - 7} )!7!}}.( {0.6} )^7 .t( {0.4})^{10 - 7}  \\    = 120 \times 0.0279936 \times 0.064 \\    = 0.215 \) Answer: The probability that exactly 7 are men is 0.215 or 21.5%.

FAQs on Binomial Distribution Formula

What is binomial distribution and binomial distribution formula in statistics.

The binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as the normal distribution. The binomial distribution, therefore, represents the probability for x successes in n trials, given a success probability p for each trial. The binomial distribution formula is for any random variable X, given by;  P(x:n,p) = n C\(_x\) p x (1-p) n-x Or P(x:n,p) = n C\(_x\) p x (q) n-x , where, n is the number of experiments, p is probability of success in a single experiment, q is probability of failure in a single experiment (= 1 – p) and takes values as 0, 1, 2, 3, 4, …, n.

What Is the Purpose of the Binomial Distribution Formula?

The binomial distribution formula allows us to compute the probability of observing a specified number of "successes" when the process is repeated a specific number of times (e.g., in a set of patients) and the outcome for a given patient is either a success or a failure.

What Is the Formula for 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 Binomial Distribution Formula for the Mean and Variance?

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

where p is the probability of success, q is the probability of failure, n = number of trials.