## Binomial Distribution |

**1. Introduction**

The binomial distribution is used when one encounters a random scenario that meets the following four requirements:

1.

**There are only two outcomes**. For example, flipping a coin (heads or tails), shooting a basket (shot made or shot not made).

2.

**There is a predetermined fixed number of observations, labeled as n**. For example, one decides to flip a coin 50 times, or shoot exactly 2 baskets.

3.

**Each observation is independent of each other observation**. For example, the outcome of each coin toss is independent of every other coin toss outcome.

4.

**Each outcome is labeled as either a "success" or "failure", and the P("success") = p and the P("failure") = (1-p) and these probabilities must be constant from observation to observation**. It's important to point out that one should label the outcome that you wish to calculate probabilities for as the "success" outcome. For example, if one wished to calculate the probability of getting exactly 4 heads when one flipped a coin 10 times, one should label getting a "head" as a "success", and getting a "tail" as a "failure".

*Note:*

1. Some books/instructors define P(Failure) as "q". In other words, 1 - p = q

**2. Binomial Random Variable**

X is a binomial random variable that counts the number of "successes" in n observations. The binomial formula requires just two parameters to be defined, they are n, the number of observations, and p, the probability of success.

One uses the following notation to define X:

**3.**

**Binomial Formula**

When one encounters a random scenario that meets the four requirements of the binomial distribution listed abov, the binomial formula below can be used to calculate the probability of getting exactly a specific number of "successes".

For example, one would use the formula below if one wanted to find the probability of flipping a fair 10 times and getting exactly 4 heads (in this case, we would label a "head" as a "success"). Using the following formula, n=10, k=4, and p=0.5 (probability of a "head" equals 0.5 for a fair coin).

**4. Mean and Standard Deviation**

A binomial random variable has the following mean and standard deviation:

**EXAMPLE 1:**

A commuter crosses a train track on his way to work each morning. The probability that he does not have to stop for a train is 90%. With the provided information:

a. Find the probability that, out of 5 days, the commuter does not have to stop for a train exactly 4 days.

b. Find the mean and standard deviation of X where X is count of days, out of 10 total days, that the commuter does not have to stop for the train.

**SOLUTION:**