What is a Binomial Random Variable?

A random variable has a binomial distribution when it counts the number of successes in a fixed number of independent trials.

Each trial must have exactly two possible outcomes, such as success/failure or heads/tails. The probability of success must also stay the same from one trial to the next.

For example, suppose we flip the same coin $n$ times. Let the probability of heads be $p$, and let $X$ count the total number of heads obtained.

Then $X$ has a binomial distribution with parameters $n$ and $p$. We write

$$X\sim\operatorname{Bin}(n,p).$$

Here:

• $n$ is the number of independent trials;

• $p$ is the fixed probability of success on each trial;

• $X$ is the total number of successes.

The possible values of $X$ are

$$0,1,2,\dots,n.$$

The binomial random variable records how many successes occur, rather than the order in which they occur.

A Simple Example

Suppose we flip a coin $5$ times, where the probability of heads is

$$p=0.6.$$

Let $X$ be the number of heads.

Then

$$X\sim\operatorname{Bin}(5,0.6).$$

One possible result is

$$H\ T\ H\ H\ T.$$

There are $3$ heads, so in this case

$$X=3.$$

Another possible result is

$$H\ H\ H\ H\ H,$$

giving

$$X=5.$$

Many different sequences of heads and tails can give the same value of $X$.

The Binomial pmf

For a binomial random variable,

$$\mathbb{P}(X=k)=\binom{n}{k}p^k(1-p)^{n-k}, \qquad k=0,1,\dots,n.$$

The term

$$p^k(1-p)^{n-k}$$

gives the probability of one particular sequence containing exactly $k$ successes and $n-k$ failures.

The binomial coefficient

$$\binom{n}{k}$$

counts how many different ways we can place those $k$ successes among the $n$ trials.

For example, suppose

$$X\sim\operatorname{Bin}(5,0.6).$$

To find the probability of exactly two successes,

$$\mathbb{P}(X=2)=\binom{5}{2}(0.6)^2(0.4)^3.$$

There are

$$\binom{5}{2}=10$$

different ways to place two successes among five trials.

Mean and Variance

For any binomial random variable,

$$\mathbb{E}(X)=np$$

and

$$\operatorname{Var}(X)=np(1-p).$$

The mean $np$ gives the expected number of successes out of the $n$ trials.

Explore the Binomial pmf

The shape of a binomial distribution changes when we change either of its two parameters, $n$ and $p$.

In the interactive graph below, the horizontal axis shows the number of successes, $k$, and each bar has height $\mathbb{P}(X=k)$.

Try changing:

• $n$, the number of trials;

• $p$, the probability of success on each trial.

Notice how the possible values and the shape of the pmf change as you change the parameters!

You can also choose a particular value of $k$ to see the corresponding probability $\mathbb{P}(X=k)$.

Binomial Distribution Summary

Background:

• Random Variables

• Probability Mass Functions

• Families of Distributions