Binomial Distribution | Brilliant Math & Science Wiki (2024)

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Alexander Katz, Geoff Pilling, Jimin Khim, and

  • Eli Ross


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
  • See Also

Formal Definition

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 \(X.\)

Finding the Binomial Distribution

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


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:

Binomial Distribution | Brilliant Math & Science Wiki (1) The binomial distribution \(b(5,0.25)\)

0.00 0.37 0.50 0.63 0.99

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? Binomial Distribution | Brilliant Math & Science Wiki (2)

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.

Properties of the Binomial Distribution

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 \(p\).

Practical Applications

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.

Binomial Test

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. \(_\square\)

See Also

  • Geometric Distribution
  • Poisson Distribution

Cite as: Binomial Distribution. Retrieved from

Binomial Distribution | Brilliant Math & Science Wiki (2024)


What is binomial distribution in math? ›

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.

What are the 4 properties of the binomial distribution? ›

1: The number of observations n is fixed. 2: Each observation is independent. 3: Each observation represents one of two outcomes ("success" or "failure"). 4: The probability of "success" p is the same for each outcome.

What is the binomial theorem in math and science? ›

The Binomial theorem tells us how to expand expressions of the form (a+b)ⁿ, for example, (x+y)⁷. The larger the power is, the harder it is to expand expressions like this directly.

What is an example of a binomial distribution in real life? ›

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.

How is the binomial theorem used in real life? ›

The impact of the economy can be calculated using the binomial mathematical theorem, and a real-life example is the US economy, which bases a substantial portion of its operations on probabilistic analysis. The binomial theorem can be used to forecast how a country's economy will perform in the near future.

What does k stand for in binomial distribution? ›

k is the number of successes (success must be defined since it can be failure to another party), • n is the number of trials (attempts, paths, legs, or missions), • ps is the probability of success for each trial, • ∗ means multiply, and • ! is the mathematical factorial notation (e.g., 4!

Is the binomial theorem calculus? ›

The coefficients of the terms in the expansion are the binomial coefficients (kn). The theorem and its generalizations can be used to prove results and solve problems in combinatorics, algebra, calculus, and many other areas of mathematics.

What is an example of a binomial in math? ›

For example, x + 2 is a binomial, where x and 2 are two separate terms. Also, the coefficient of x is 1, the exponent of x is 1 and 2 is the constant here. Therefore, A binomial is a two-term algebraic expression that contains variable, coefficient, exponents and constant.

Why is binomial distribution important? ›

The binomial distribution model 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 a medical example of binomial distribution? ›

For example, a medical researcher may want to know the probability of a certain number of patients responding positively to a new treatment. Using binomial distribution, the researcher can calculate the probability of a certain number of successful treatments in a fixed sample of patients.

In what situations could you use a binomial distribution? ›

When using certain sampling methods, there is a possibility of having trials that are not completely independent of each other, and binomial distribution may only be used when the size of the population is large vis-a-vis the sample size. An example of independent trials may be tossing a coin or rolling a dice.

What is binomial distribution formula examples? ›

For example, let's suppose you wanted to know the probability of getting a 1 on a die roll. if you were to roll a die 20 times, the probability of rolling a one on any throw is 1/6. Roll twenty times and you have a binomial distribution of (n=20, p=1/6).

What does the mean of a binomial distribution tell us? ›

The mean is the expected number of successes in the experiment. The standard deviation shows the spread of the probability - meaning that any number of successes not in between and would be unusual.

How to solve a binomial? ›

To solve a binomial problem, if your x term is being multiplied by a number, you'll divide both sides of your equation by that number. If your x term is being divided by a number, you'll multiply both sides of your equation by that number.

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