# 7.4 - Binomial Probabilities

Binomial probabilities are covered in section P.4 of the Lock^5 textbook.

With proportions, when the sample size is small it is inappropriate to approximate the sampling distribution with a normal distribution. In those cases a binomial distribution may be used to approximate the sampling distribution.

**Binomial random variable**: A specific type of discrete random variable that counts how often a particular event occurs in a fixed number of tries or trials

For a variable to be a **binomial random variable**, **ALL** of the following conditions must be met:

- There are a fixed number of trials (a fixed sample size)
- The probability of a success is the same on each trial
- Trials are independent of one another

### Examples of Binomial Random Variables

- Number of correct guesses at 30 true-false questions when you randomly guess all answers
- Number of winning lottery tickets when you buy 10 tickets of the same kind
- Number of left-handers in a randomly selected sample of 100 unrelated people
- Number of tails when flipping a coin 10 times

**Notation **

n= number of trials

p= probability event of interest occurs on any one trial

### Example: True-False Test

Number of correct guesses at 30 true-false questions when you randomly guess all answers.

There are 30 trials, therefore *n* = 30.

There are two possible outcomes (true and false) that are equally probable, therefore \(p = \frac{1}{2} = 0.5\).