Module 4: Discrete Random Variables
Geometric Distribution
Barbara Illowsky & OpenStax et al.
There are three main characteristics of a geometric experiment.
- There are one or more Bernoulli trials with all failures except the last one, which is a success. 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 bullseye until you hit the bullseye. The first time you hit the bullseye is a “success” so you stop throwing the dart. It might take six tries until you hit the bullseye. 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 is the same for each trial. [latex]p+q=1[/latex] and [latex]q=1−p[/latex]. For example, the probability of rolling a three when you throw one fair die is [latex]frac{1}{6}[/latex], the probability of a failure. The probability of getting a three on the fifth roll is [latex]X=[/latex] the number of independent trials until the first success.
Example
You play a game of chance that you can either win or lose (there are no other possibilities) until you lose. Your probability of losing is [latex]p=0.57[/latex]. What is the probability that it takes five games until you lose?
[reveal-answer q=”416195″]Show Solution[/reveal-answer]
[hidden-answer a=”416195″]Let [latex]X=[/latex] the number of games you play until you lose (includes the losing game). Then X takes on the values 1, 2, 3, … (could go on indefinitely). The probability question is [latex]P(x=5)[/latex].
[/hidden-answer]
Try It
You throw darts at a board until you hit the center area. Your probability of hitting the center area is [latex]p=0.17[/latex]. You want to find the probability that it takes eight throws until you hit the center. What values does X take on?
[reveal-answer q=”129956″]Show Solution[/reveal-answer]
[hidden-answer a=”129956″]1, 2, 3, 4, … n. It can go on indefinitely.[/hidden-answer]
Example
A safety engineer feels that 35% 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. On average, how many reports would the safety engineer expect to look at until she finds a report showing an accident caused by employee failure to follow instructions? What is 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?
[reveal-answer q=”885549″]Show Solution[/reveal-answer]
[hidden-answer a=”885549″]Let [latex]X=[/latex] the number of accidents the safety engineer must examine until she finds a report showing an accident cause