Geometric Distribution

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]

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 caused by employee failure to follow instructions. X takes on the values 1, 2, 3, …. The first question asks you to find the expected value or the mean. The second question asks you to find [latex]P(xgeq3)[/latex]. (“At least” translates to a “greater than or equal to” symbol).[/hidden-answer]

Example

Suppose that you are looking for a student at your college who lives within five miles of you. You know that 55% 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).

1. Let [latex]X=[/latex] the number of ____________ you must ask ____________ one says yes.
2. What values does X take on?
3. What are p and q?
4. The probability question is P(_______).

[reveal-answer q=”517250″]Show Solution[/reveal-answer]
[hidden-answer a=”517250″]

1. Let [latex]X=[/latex] the number of students you must ask until one says yes.
2. 1, 2, 3, …, (total number of students)
3. [latex]p=0.55[/latex]; [latex]q=0.45[/latex]
4. [latex]P(x=4)[/latex]

[/hidden-answer]

Example

Assume that the probability of a defective computer component is 0.02. Components are randomly selected. Find the probability that the first defect is caused by the seventh component tested. How many components do you expect to test until one is found to be defective?

Let [latex]X=[/latex] the number of computer components tested until the first defect is found.

X takes on the values 1, 2, 3, … where [latex]p = 0.02[/latex].

[latex]X{sim}G(0.02)[/latex]

Find [latex]P(x=7)[/latex]. [latex]P(x=7)=0.0177[/latex].

To find the probability that [latex]x=7[/latex],

• Enter 2nd, DISTR
• Scroll down and select geometpdf(
• Press ENTER
• Enter 0.02, 7); press ENTER to see the result: [latex]P(x=7)=0.0177[/latex]

To find the probability that [latex]xleq7[/latex], follow the same instructions EXCEPT select E:geometcdf(as the distribution function.

The probability that the seventh component is the first defect is 0.0177.

The graph of [latex]X{sim}G(0.02)[/latex] is:

The y-axis contains the probability of x, where [latex]X=[/latex] the number of computer components tested.

The number of components that you would expect to test until you find the first defective one is the mean, [latex]displaystyle{mu}={50}[/latex].

The formula for the mean is [latex]displaystyle{mu}=frac{{1}}{{p}}=frac{{1}}{{0.02}}={50}[/latex]

The formula for the variance is [latex]displaystyle{sigma}^{{2}}={(frac{{1}}{{p}})}{(frac{{1}}{{p}}-{1})}={(frac{{1}}{{0.02}})}{(frac{{1}}{{0.02}}-{1})}={2},{450}[/latex]

The standard deviation is [latex]displaystyle{sigma}=sqrt{{{(frac{{1}}{{p}})}{(frac{{1}}{{p}}-{1})}}}=sqrt{{{(frac{{1}}{{0.02}})}{(frac{{1}}{{0.02}}-{1})}}}={49.5}[/latex]

Example

The lifetime risk of developing pancreatic cancer is about one in 78 (1.28%). Let [latex]X=[/latex]the number of people you ask until one says he or she has pancreatic cancer. Then X is a discrete random variable with a geometric distribution: [latex]displaystyle{X}~{G}{(frac{{1}}{{78}})}{quadtext{or}quad}{X}~{G}{({0.0128})}[/latex]

1. What is the probability of that you ask ten people before one says he or she has pancreatic cancer?
2. What is the probability that you must ask 20 people?
3. Find the (i) mean and (ii) standard deviation of X.

[reveal-answer q=”556559″]Show Solution[/reveal-answer]
[hidden-answer a=”556559″]

1. [latex]P(x=10)=text{geometpdf}(0.0128,10)=0.0114[/latex]
2. [latex]P(x=20)=text{geometpdf}(0.0128,20)=0.01[/latex]
3. 1. [latex]text{Mean}={mu}=frac{{1}}{{p}}=frac{{1}}{{0.0128}}={78}[/latex]
   2. [latex]text{Standard Deviation}={sigma}=sqrt{{frac{{{1}-{p}}}{{{p}^{{2}}}}}}=sqrt{{frac{{{1}-{0.0128}}}{{0.0128}^{{2}}}}}approx{77.6234}[/latex]

[/hidden-answer]