# 7.4 - Binomial Probabilities

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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:

1. There are a fixed number of trials (a fixed sample size)
2. The probability of a success is the same on each trial
3. 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 $$p = \frac{1}{2} = 0.5$$.