### Introduction

There are three main characteristics of a **geometric experiment**.

- Repeating independent Bernoulli trials until a success is obtained. Recall that a Bernoulli trial is a binomial experiment with number of trials
*n*= 1. In other words, you keep repeating what you are doing until the first success. Then you stop. For example, you throw a dart at a bull's-eye until you hit the bull's-eye. The first time you hit the bull's-eye is a*success*, so you stop throwing the dart. It might take six tries until you hit the bull's-eye. You can think of the trials as failure, failure, failure, failure, failure, success,**stop**. - In theory, the number of trials could go on forever. There must be at least one trial.
- The probability,
*p*, of a success and the probability,*q*, of a failure do not change from trial to trial.*p*+*q*= 1 and*q*= 1 −*p*. For example, the probability of rolling a three when you throw one fair die is $\frac{1}{6}$. This is true no matter how many times you roll the die. Suppose you want to know the probability of getting the first three on the fifth roll. On rolls one through four, you do not get a face with a three. The probability for each of the rolls is*q*= $\frac{\text{5}}{\text{6}}$, the probability of a failure. The probability of getting a three on the fifth roll is $\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)\left(\frac{1}{6}\right)$ = .0804.

*X* = the number of independent trials until the first success.

*p* = the probability of a success, *q* = 1 – *p* = the probability of a failure.

There are shortcut formulas for calculating mean *μ*, variance *σ*^{2}, and standard deviation *σ* of a geometric probability distribution. The formulas are given as below. The deriving of these formulas will not be discussed in this book.

### Example 4.16

Suppose a game has two outcomes, win or lose. You repeatedly play that game **until** you lose. The probability of losing is *p* = 0.57.

If we let *X* = the number of games you play until you lose (includes the losing game), then *X* is a geometric random variable. All three characteristics are met. Each game you play is a Bernoulli trial, either win or lose. You would need to play at least one game before you stop. *X* takes on the values 1, 2, 3, . . . (could go on indefinitely). Since we are measuring the number of games you play until you lose, we define a success as losing a game and a failure as winning a game. The probability of a success $p=.57$ and the probability of a failure *q*= 1-*p* = 1–0.57 = 0.43. Both *p* and *q* remain the same from game to game.

If we want to find the probability that it takes five games until you lose, then the probability could be written as *P*(*x* = 5). We will explain how to find a geometric probability later in this section.

You throw darts at a board until you hit the center area. Your probability of hitting the center area is *p* = 0.17. You want to find the probability that it takes eight throws until you hit the center. What values does *X* take on?

### Example 4.17

A safety engineer feels that 35 percent of all industrial accidents in her plant are caused by failure of employees to follow instructions. She decides to look at the accident reports (selected randomly and replaced in the pile after reading) *until* she finds one that shows an accident caused by failure of employees to follow instructions.

If we let *X* = the number of accidents the safety engineer must examine until she finds a report showing an accident caused by employee failure to follow instructions, then *X* is a geometric random variable. All three characteristics are met. Each accident report she reads is a Bernoulli trial: the accident was either caused by failure of employees to follow instructions or not. She would need to read at least one accident report before she stops. *X* takes on the values 1, 2, 3, . . . (could go on indefinitely). Since we are measuring the number of reports she needs to read until one that shows an accident caused by failure of employees to follow instructions, we define a success as an accident caused by failure of employees to follow instructions. If an accident was caused by another reason, the report is defined as a failure. The probability of a success *p* = .35 and the probability of a failure $q=\mathrm{}1-p=1-.35=.65$. Both *p* and *q* remain the same from report to report.

If we want to find the probability that the safety engineer will have to examine at least three reports until she finds a report showing an accident caused by employee failure to follow instructions, then the probability could be written as $p=.35$. If we want to find how many reports, on average, the safety engineer would *expect* to look at until she finds a report showing an accident caused by employee failure to follow instructions, we need to find the expected value *E*(*x*). We will explain how to solve these questions later in this section.

An instructor feels that 15 percent of students get below a C on their final exam. She decides to look at final exams (selected randomly and replaced in the pile after reading) until she finds one that shows a grade below a C. We want to know the probability that the instructor will have to examine at least 10 exams until she finds one with a grade below a C. What is the probability question stated mathematically?

### Example 4.18

Suppose that you are looking for a student at your college who lives within five miles of you. You know that 55 percent of the 25,000 students do live within five miles of you. You randomly contact students from the college *until* one says he or she lives within five miles of you. What is the probability that you need to contact four people?

This is a geometric problem because you may have a number of failures before you have the one success you desire. Also, the probability of a success stays the same each time you ask a student if he or she lives within five miles of you. There is no definite number of trials (number of times you ask a student).

a. Let *X* = the number of ________ you must ask ________ one says yes.

a. Let *X* = the number of *students* you must ask *until* one says yes.

b. What values does *X* take on?

b. 1, 2, 3, . . ., (total number of students)

c. What are *p* and *q*?

c. *p* = .5, *q* = .45

d. The probability question is *P*(_______).

d. *P*(*x* = 4)

You need to find a store that carries a special printer ink. You know that of the stores that carry printer ink, 10 percent of them carry the special ink. You randomly call each store until one has the ink you need. What are *p* and *q*?