# Day 4: Binomial Distribution I

# Day 4: Binomial Distribution I

Terms you'll find helpful in completing today's challenge are outlined below.

### Random Variable

A *random variable*, , is the real-valued function in which there is an event for each interval where . You can think of it as the set of probabilities for the possible outcomes of a sample space. For example, if you consider the possible sums for the values rolled by four-sided dice:

**Note:** When we roll two dice, the value rolled by each die is independent of the other.

### Binomial Experiment

A *binomial experiment* (or *Bernoulli trial*) is a statistical experiment that has the following properties:

- The experiment consists of repeated trials.
- The trials are independent.
- The outcome of each trial is either
*success*() or*failure*().

### Bernoulli Random Variable and Distribution

The sample space of a binomial experiment only contains two points, and . We define a *Bernoulli random variable* to be the random variable defined by and . If we consider the probability of success to be and the probability of failure to be (where ), then the probability mass function (*PMF*) of is:

We can also express this as:

### Binomial Distribution

We define a *binomial process* to be a binomial experiment meeting the following conditions:

- The number of successes is .
- The total number of trials is .
- The probability of success of trial is .
- The probability of failure of trial , where .
- is the
*binomial probability*, meaning the probability of having exactly successes out of trials.

The *binomial random variable* is the number of successes, , out of trials.

The *binomial distribution* is the probability distribution for the binomial random variable, given by the following probability mass function:

**Note:** Recall that . For further review, see the Combinations and Permutations Tutorial.

### Cumulative Probability

We consider the distribution function for some real-valued random variable, , to be . Because this is a non-decreasing function that accumulates all the probabilities for the values of up to (and including) , we call it the *cumulative distribution function* (*CDF*) of . As the CDF expresses a cumulative range of values, we can use the following formula to find the cumulative probabilities for all :

### Example

A fair coin is tossed times. Find the following probabilities:

- Getting heads.
- Getting at least heads.
- Getting at most heads.

For this experiment, , , and . The respective probabilities for the above three events are as follows:

- The probability of getting heads is:
- The probability of getting
*at least*heads is: - The probability of getting
*at most*heads is: