3.4 The Binomial Probability Distribution

Contents

• 3.4.1 The Binomial Random Variable and Distribution
• 3.4.2 Using Binomial Tables
• 3.4.3 The Mean and Variance of X
• EXERCISES Section 3.4 (46-67)

Introduction

Many experiments conform to the following requirements (either exactly or approximately):

1. The experiment consists of a sequence of $n$ smaller experiments (called trials):
   • $n$ is fixed in advance
2. Each trial results in one of two possible outcomes (dichotomous trials):
   • Denote outcomes as success $(S)$ and failure $(F)$
   • Assignment of $S$ and $F$ is arbitrary
3. The trials are independent:
   • The outcome of any particular trial does not influence the outcome of other trials
4. The probability of success $P(S)$ is constant from trial to trial:
   • Denote this probability by $p$

binomial experiment

An experiment for which Conditions 1-4 (a fixed number of dichotomous, independent, homogenous trials) are satisfied is called a binomial experiment.

• EX 3.27 examples of binomial experiment
• EX 3.28 color of pea seed
• EX 3.29 juror pool

We will use the following rule of thumb in deciding whether a “without-replacement” experiment can be treated as being binomial.

Remark

• Many experiments involve a sequence of independent trials
• More than two possible outcomes on any one trial
• A binomial experiment can be created by:
  • Dividing the possible outcomes into two groups
• Rule of thumb for treating a “without-replacement” experiment as binomial:
  • Consider sampling without replacement from a dichotomous population of size $N$
    • If the sample size $n$ is at most 5% of the population size,
  • The experiment can be analyzed as though it were a binomial experiment

By “analyzed”, we mean that probabilities based on the binomial experiment assumptions will be quite close to the actual “without-replacement” probabilities, which are typically more difficult to calculate.

• In the first scenario of EX 3.29 juror pool,
  • $n/N=6/50=.12>.05$ ,
  • so the binomial experiment is not a good approximation,
• but in the second scenario,
  • $n/N=10/500,000\ll .05$ .