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A random variable follows a **Bernoulli distribution** if it only has two possible outcomes: 0 or 1.

For example, suppose we flip a coin one time. Let the probability that it lands on heads be *p*. This means the probability that it lands on tails is 1-*p*.

Thus, we could write:

In this case, random variable *X* follows a Bernoulli distribution. It can only take on two possible values.

**Now, if we flip a coin multiple times then the sum of the Bernoulli random variables will follow a Binomial distribution.**

For example, suppose we flip a coin 5 times and we want to know the probability of obtaining heads *k *times. We would say that the random variable *X* follows a Binomial distribution.

If a random variable *X* follows a Binomial distribution, then the probability that *X* = *k* successes can be found by the following formula:

**P(X=k) = _{n}C_{k} * p^{k} * (1-p)^{n-k}**

where:

**n:**number of trials**k:**number of successes**p:**probability of success on a given trialthe number of ways to obtain_{n}C_{k}:*k*successes in*n*trials

For example, suppose we flip a coin 3 times. We can use the formula above to determine the probability of obtaining 0 heads during these 3 flips:

P(X=0) = _{3}C_{0} * .5^{0} * (1-.5)^{3-0} = 1 * 1 * (.5)^{3} = **0.125**

**When n = 1 trial, the Binomial distribution is equivalent to the Bernoulli distribution.**

**Important Notes**

Here are a couple important notes in regards to the Bernoulli and Binomial distribution:

**1. A random variables that follows a Bernoulli distribution can only take on two possible values, but a random variable that follows a Binomial distribution can take on several values.**

For example, in a single coin flip we will either have 0 or 1 heads. However, in a series of 5 coin flips we could have 0, 1, 2, 3, 4, or 5 heads.

**2. In order for a random variable to follow a Binomial distribution, the probability of “success” in each Bernoulli trial must be equal and independent.**

For example, if we define “success” as landing on heads, then the probability of success on each coin flip is equal to 0.5 and each flip is independent – the outcome of one coin flip does not affect the outcome of another.

**Additional Resources**

An Introduction to Binomial Experiments

An Introduction to the Binomial Distribution

Understanding the Shape of a Binomial Distribution