# Bernoulli & Binomial Random Variables

In DISCOVERY, we will talk about specific types of random variables, such as Bernoulli and Binomial random variables.

## Bernoulli Distribution for Discrete Random Variables

Any event that has exactly two outcomes with a fixed probability is called a Bernoulli random variable. Every Bernoulli distribution has a probability, *p*, describing the probability of that event occurring. We know a lot of these already!

- For example: If we are looking at the event of drawing a queen from a deck of 52 cards, we know that the probability of doing this,
*p*, is 4/52 or 1/13.

We can say this this is a Bernoulli Distribution, with p= 1/13. Oftentimes this is written as *X*~Bernoulli(p=1/13). The means that the random variable *X* is a Bernoulli random variable with a probability of success = 1/13.

The idea behind the Bernoulli distribution is that the event is repeated only once. But what happens if we repeat the event more than once, assuming that we are dealing with independent events?

This is where the Binomial Distribution comes into play!

## Binomial Distribution for Discrete Random Variables

The Binomial Distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent events. In other words, the Binomial Distribution is the sum of *n* independent Bernoulli random variables.

Just like a Bernoulli random variable, random variables that follows the binomial distribution can only take on two outcomes: success or failure (1 or 0). Recall that a single success/failure experiment is called a Bernoulli.

**In other words, for a single trial (n=1), the binomial distribution is a Bernoulli distribution.**

We can easily calculate the Expected Value and Standard Error for Binomial Random Variables. For all binomial distributions B(n, p) where n is the number of trials and p is the probability of success:

# Example Walk-Throughs with Worksheets

### Video 1: Bernoulli Examples

### Video 2: Binomial Examples

# Practice Questions

**Q1**: If there are 3 red marbles, 4 blue marbles, and 2 yellow marbles in a bag and we look at the event of drawing a blue marble (by only drawing one marble), would it be a Bernoulli or Binomial distribution?

**Q2**: Which of the following is not a characteristic of a Binomial distribution?

**Q3**: How do we write a Bernoulli distribution with a probability of 2/9?

**Q4**: What is the expected value for a Binomial Random Variable with 3 trials and a 1/3 probability?

**Q5**: True or false: for a single trial, the Binomial distribution is a Bernoulli distribution.