Binomial Probabilities

A sequence of n observations is called a binomial process  if
(i) each observation results in one of two possible outcomes (which we call success  and failure )
(ii) the probability of success is p and the probability of failure is q=1-p for all observations
(iii) the observations are independent of each other.

$$\begin{array}{ccccccccccc} \multicolumn{2}{c}{\mbox{Obs 1}} ... ... \searrow q \\ S & F & & S & F & & S & F & & S & F \end{array}$$

Let X denote the total number of successes among the n observations. Then X is called a binomial random variable  with parameters n and p.

The following are all binomial random variables.

Example 1

A Stat 216 multiple choice quiz has 5 questions, with each question having 4 choices. Let X be the number of correct answers (C) by someone who is guessing on all questions. Then X is a Binomial random variable with parameter values n=5 and p=1/4.

$$\begin{array}{cccccccccccccc} \multicolumn{2}{c}{\mbox{Ques ... ...C & W & & C & W & & C & W & & C & W & & C & W \\ \end{array}$$

Example 2

Available data shows that 40% of telephone respondents agree to be interviewed for market research surveys. Suppose that the polling organization Reliable Research randomly selects and dials telephone numbers until 50 respondents are reached. Let X be the number of respondents (out of the 50) who agree to be interviewed. Then X is a Binomial random variable with parameter values n=50 and p=.40.

Example 3

Historically, 20% of television buyers at TV World purchase the store's extended warranty. Suppose that 300 TV sets were sold during the previous quarter. Let X be the number of extended warranties that were sold along with the 300 sets. Then X is a Binomial random variable with parameter values n=300 and p=.20.