# Binomial Distribution

An experiment is modeled on the binomial distribution if:

1. In each trial of the experiment there are only two possible outcomes: the event **A** (success) and its opposite .

2.The probability of event **A** is a constant that does not vary from one test to another. It is denoted by **p**.

3.The probability of is **1 − p**.

4.The outcome obtained in each trial is independent of the previous outcomes.

#### Random Binomial Variable

The binomial variable is a discreet random variable that can only take the values 0, 1, 2, 3, 4, ..., n assuming that n trials have been conducted.

#### Example

k = 6, when flipping a coin 10 times and 6 heads are obtained.

## Binomial Distribution

The **binomial distribution** is usually expressed by **B(n, p)**.

**n** is the number of trials

**k** is the number of successes.

**p** is the probability of success.

#### Example

The latest novel by an author has been very successful to the point that 80% of avid readers have already read it.

1. What is the probability that within a group of four avid readers, exactly two of them have already read the novel?

n = 4

p = 0.8

1 − p = 0.2

B(4, 0.8)

2.What about a maximum of two people?

**Expected Value **

**Variance **

#### Standard Deviation

#### Example

The probability of a defective item produced by a factory is 0.02. If a shipment of 10,000 items are sent to a store, determine the expected number of defective items, the variance and standard deviation for the shipment.