Chapter 4: Discrete Random Variables

4.8 Using Technology for the Geometric Distribution

Learning Objectives

By the end of this section, the student should be able to:

  • use technology to solve problems involving the Geometric Distribution

This is another section to show technology tools that can be used to help with the statistics concepts taught in this chapter. This section will cover tools to help solve problems involving the Geometric Distribution.

Graphing Calculator

Similar to all the other technology sections, the main tool introduced will be the TI-83, 83+, 84, or 84+ Graphing Calculator. The graphing calculator can help to find the probabilities for the geometric probability distribution.

 

Using the TI-83, 83+, 84, 84+ Calculator for Geometric Distributions

  1. Access [DISTR] by pressing 2nd key, vars key. (Note: DISTR stands for "Distributions")
  2. The functions for the Geometric Distribution are:
    • <geometpdf()> corresponds to P(X = x)
    • <geometcdf()> corresponds to P(X ≤ x)

The syntax for the instructions is as follows:

  • geometpdf(p, value) to calculate P(X = value).
  • geometcdf(p, value) to calculate P(X ≤ value).

Legend

  • A blank calculator button represents a button press
  • [ ] represents yellow command or green letter behind a key
  • < > represents items on the screen

 

Let's use these calculator functions with some of the same examples we did in 4.4 Geometric Distribution.

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.

Let [latex]X =[/latex] the number of computer components tested until the first defect is found; [latex]X[/latex] takes on the values 1, 2, 3, ... where p = 0.02. We write [latex]X \sim G(0.02[/latex].

  1. Find the probability that the first defect is found in the seventh component tested.
  2. Find the probability that the first defect is found in less than nine of the components tested.

For this example, to find the probability that the first defect is found in the seventh component tested, we are looking for [latex]P(X = 7)[/latex]. To find the probability that [latex]x = 7[/latex] using a calculator, use [latex]\text{geometpdf}(0.02, 7)[\latex]. The result is [latex]P(X = 7) = 0.0177[/latex].

With the second part, the probability that the first defect is found in less than nine of the components tested, we are looking for [latex]P(X \leq 8)[/latex]. Remember, this is a discrete random variable, so [latex]x < 9[/latex] is [latex]x \le 8[/latex]. To find the probability that [latex]x \leq 8[/latex] using a calculator, use [latex]\text{geometcdf}(0.02, 8)[\latex]. The result is [latex]P(X \leq 8) = 0.1492[/latex].

 

Note

To compute geometric probabilities on a graphing calculator, the syntax is 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(p, x).

 

Here's some more examples, with solution, for you to try, from the problems covered in the Section.

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. 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?

Solution

The question asks you to find [latex]P(x \ge 3)[/latex]. ("At least" translates to a "greater than or equal to" symbol.) Thinking about how to use the geometcdf() distribution and what we know about discrete random variables, [latex]P(X \ge 3) = 1 - P(X \le 2)[/latex]. So, we can use our distribution in the calculator by doing [latex]1 - \text{geometcdf}(.35, 2) = 0.4225[\latex].

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. Solution [latex]P(x = 9) = \text{geometpdf}(0.01, 9) = 0.0092[/latex]

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 [latex]X[/latex]?

  1. What is the probability that you ask five women before one says she is literate?
  2. What is the probability that you must ask ten women?

Solution

[latex]X \sim G(0.12)[/latex]

  1. [latex]P(x = 5) = \text{geometpdf}(0.12, 5) = 0.0720[/latex]
  2. [latex]P(x = 10) = \text{geometpdf}(0.12, 10) = 0.0380[/latex]

Your Turn!

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 [latex]X[/latex] is a discrete random variable with a geometric distribution: [latex]X \sim G(0.0128)[/latex].

  1. What is the probability that you ask 10 people before one says he or she has pancreatic cancer?
  2. What is the probability that you must ask 20 people?

Solution

  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]

 

Helpful Videos for the Graphing Calculator

Below are links to helpful videos for using the graphing calculator for the concepts covered on this page:

 

Additional Technology Tools

In addition to the graphing calculator, there are some additional technology tools that can be used for the concepts covered on this page.

Microsoft Excel

In Microsoft 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().

 

Section Practice

The World Bank records the prevalence of HIV in countries around the world. According to their data, “Prevalence of HIV refers to the percentage of people ages 15 to 49 who are infected with HIV.” In South Africa, the prevalence of HIV is 17.3%. Let [latex]X =[/latex] the number of people you test until you find a person infected with HIV.

  1. Sketch a graph of the distribution of the discrete random variable [latex]X[/latex].
  2. What is the probability that you must test 30 people to find one with HIV?
  3. What is the probability that you must ask ten people?
  4. Find the (i) mean and (ii) standard deviation of the distribution of [latex]X[/latex].
Solution
  1. [latex]X \sim G(0.173)[/latex]
  2. [latex]P(x = 30) = \text{geometpdf} (0.173, 30) = 0.0007[/latex]
  3. [latex]P(x = 10) = \text{geometpdf} (0.173, 10) = 0.0313[/latex]
  4. (i) [latex]\text{Mean} = \mu = \frac{1}{p}=\frac{1}{0.173} \approx{5.7804}[/latex]
    (ii) [latex]\text{Standard Deviation} = \sigma =\frac{\sqrt{(1−p)}}{p}=\frac{\sqrt{1−0.173}}{0.173}= 5.2566[/latex]

According to a recent Pew Research poll, 75% of millennials (people born between 1981 and 1995) have a profile on a social networking site. Let [latex]X =[/latex] the number of millennials you ask until you find a person without a profile on a social networking site.

  1. Describe the distribution of [latex]X[/latex].
  2. Find the (i) mean and (ii) standard deviation of [latex]X[/latex].
  3. What is the probability that you must ask ten people to find one person without a social networking site?
  4. What is the probability that you must ask 20 people to find one person without a social networking site?
  5. What is the probability that you must ask at most five people?
Solution
  1. [latex]X \sim G(0.25)[/latex]
  2. (i) [latex]\text{Mean} = \mu = \frac{1}{p} = \frac{1}{0.25} =4[/latex]
    (ii) [latex]\text{Standard Deviation} = \sigma = \sqrt{\frac{1-p}{{p}^{2}}} = \sqrt{\frac{1-\text{0}\text{.25}}{{0.25}^{2}}} \approx 3.4641[/latex]
  3. [latex]P(x = 10) = \text{geometpdf} (0.25, 10) = 0.0188[/latex]
  4. [latex]P(x = 20) = \text{geometpdf} (0.25, 20) = 0.0011[/latex]
  5. [latex]P(x \le 5) = \text{geometcdf} (0.25, 5) = 0.7627[/latex]

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