Chapter 4: Discrete Random Variables
4.4 Geometric Distribution
Learning Objectives
By the end of this section, you should be able to:
- Identify the components of a geometric experiment
- Use the formulas for a geometric random variable to compute mean, variance, and standard deviation
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 [latex]p[/latex] of a success and the probability [latex]q[/latex] of a failure is the same for each trial with [latex]p+q = 1[/latex], so [latex]q = 1-p[/latex].
The geometric random variable [latex]X =[/latex] the number of independent trials until the first success. The probability that [latex]X[/latex] equals a particular value [latex]x[/latex], that is, the probability we first succeed on a particular trial, is
[latex]P(X = x) = q^{(x-1)} p.[/latex]
Example
Suppose that when rolling a fair die, you want to know the probability of getting the first three on the fifth roll.
The experiment here is to roll a fair die until you get a three, then stop. This is a geometric experiment because the probability of rolling a three when you throw one fair die is [latex]\frac{1}{6}[/latex] no matter how many times you roll the die, and you either get a 3 (success) or get one of the other 5 numbers (failure).
So for the event to occur, on rolls one through four, you do not get a face with a three. The probability for each of the rolls is [latex]q = \frac{\text{5}}{\text{6}}[/latex], the probability of a failure. The probability of getting a three on the fifth roll is [latex]\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)[/latex] = 0.0804
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?
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. 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 P(x ≥ 3). (“At least” translates to a “greater than or equal to” symbol.)
Your Turn!
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 p = 0.57. What is the probability that it takes five games until you lose?
Your Turn!
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 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.
Solution
a. Let X = the number of students you must ask until one says yes.
b. What values does X take on?
Solution
b. 1, 2, 3, …, (total number of students)
c. What are p and q?
Solution
c. p = 0.55; q = 0.45
d. The probability question is P(_______).
Solution
d. P(x = 4)
Expected Value and Standard Deviation
The notation for the Geometric Probability Distribution function is [latex]G[/latex], and we denote “X is a random variable with a geometric distribution” as [latex]X \sim G(p)[/latex].
The parameter is [latex]p[/latex], the probability of a success for each trial.
The formula for the mean is [latex]\mu = \frac{1}{p}.[/latex]
The formula for the variance is [latex]\sigma^2 = \left(\frac{1}{p}\right)\left(\frac{1}{p}-1\right)[/latex].
The standard deviation is [latex]\sigma = \sqrt{\left(\frac{1}{p}\right)\left(\frac{1}{p}-1\right)}[/latex].
Note
To compute geometric probabilities on a graphing calculator, go into 2nd DISTR. The syntax for the instructions are as follows:
To calculate [latex]P(X = x)[/latex]: geometpdf(p, x).
To calculate [latex]P(X \leq x)[/latex]: geometcdf(p, x).
If you wanted to instead find [latex]P(X>x)[/latex], use 1 – geometcdf([latex]p,x[/latex]).
In Excel, both geometric probabilities are computed using NEGBINOM.DIST(x-1, 1, p, True/False), where False computes [latex]P(X=x)[/latex] and is equivalent to geometpdf, and True computes [latex]P(X \leq x)[/latex] and is equivalent to geometcdf.
Example
Suppose that at a particular computer component manufacturing facility that the probability of a defective computer component is 0.02. For quality assurance, components are randomly selected. Find the probability that the first defect is found in the seventh component tested. How many components do you expect to test until one is found to be defective?
Let X = the number of computer components tested until the first defect is found; X takes on the values 1, 2, 3, … where p = 0.02. We write [latex]X \sim G(0.02[/latex].
We are looking for [latex]P(X=7)[/latex]. To find the probability that x = 7 using a calculator,
- 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 [latex]P(X \leq 7)[/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 below.
The y-axis contains the probability of [latex]x[/latex] 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]\mu = \frac{1}{p}[/latex] = [latex]\frac{1}{0.02}[/latex] = 50.
The variance is [latex]\sigma^2 = \left(\frac{1}{p}\right)\left(\frac{1}{p}-1\right)[/latex] = [latex]\left(\frac{1}{0.02}\right)\left(\frac{1}{0.02}-1\right)[/latex] = 2,450.
The standard deviation is σ = [latex]\sqrt{\left(\frac{1}{p}\right)\left(\frac{1}{p}-1\right)}[/latex] = [latex]\sqrt{\left(\frac{1}{0.\text{02}}\right)\left(\frac{1}{0.\text{02}}-1\right)}[/latex] = 49.5.
Your Turn!
The probability of a defective steel rod is 0.01. Steel rods are selected at random. Find the probability that the first defect occurs on the ninth steel rod. Use a calculator or computer to find the answer.
Your Turn!
The literacy rate for a nation measures the proportion of people age 15 and over who can read and write. The literacy rate for women in Afghanistan is 12%. Let [latex]X=[/latex] the number of Afghani women you ask until one says that she is literate.
- What is the probability distribution of X?
- What is the probability that you ask five women before one says she is literate?
- What is the probability that you must ask ten women?
- Find the (i) mean and (ii) standard deviation of X.
References
“Millennials: A Portrait of Generation Next,” PewResearchCenter. Available online at http://www.pewsocialtrends.org/files/2010/10/millennials-confident-connected-open-to-change.pdf (accessed May 15, 2013).
“Millennials: Confident. Connected. Open to Change.” Executive Summary by PewResearch Social & Demographic Trends, 2013. Available online at http://www.pewsocialtrends.org/2010/02/24/millennials-confident-connected-open-to-change/ (accessed May 15, 2013).
“Prevalence of HIV, total (% of populations ages 15-49),” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/SH.DYN.AIDS.ZS?order=wbapi_data_value_2011+wbapi_data_value+wbapi_data_value-last&sort=desc (accessed May 15, 2013).
Pryor, John H., Linda DeAngelo, Laura Palucki Blake, Sylvia Hurtado, Serge Tran. The American Freshman: National Norms Fall 2011. Los Angeles: Cooperative Institutional Research Program at the Higher Education Research Institute at UCLA, 2011. Also available online at http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf (accessed May 15, 2013).
“Summary of the National Risk and Vulnerability Assessment 2007/8: A profile of Afghanistan,” The European Union and ICON-Institute. Available online at http://ec.europa.eu/europeaid/where/asia/documents/afgh_brochure_summary_en.pdf (accessed May 15, 2013).
“The World FactBook,” Central Intelligence Agency. Available online at https://www.cia.gov/library/publications/the-world-factbook/geos/af.html (accessed May 15, 2013).
“UNICEF reports on Female Literacy Centers in Afghanistan established to teach women and girls basic resading [sic] and writing skills,” UNICEF Television. Video available online at http://www.unicefusa.org/assets/video/afghan-female-literacy-centers.html (accessed May 15, 2013).
a statistical experiment with the following properties:
1. There are one or more Bernoulli trials with all failures except the last one, which is a success.
2. In theory, the number of trials could go on forever. There must be at least one trial.
3. The probability, p, of a success and the probability, q, of a failure do not change from trial to trial.
a discrete random variable (RV) that arises from the Bernoulli trials; the trials are repeated until the first success. The geometric variable X is defined as the number of trials until the first success. Notation: X ~ G(p). The mean is μ = [latex]\frac{1}{p}[/latex] and the standard deviation is σ = [latex]\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}[/latex]. The probability of exactly x failures before the first success is given by the formula: P(X = x) = p(1 – p)^(x – 1)