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.

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.

**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.