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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).

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 = 0.55; q = 0.45

d. The probability question is P (_______).

d. P ( X = 4)

Try it

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

p = 0.1

q = 0.9

Notation for the geometric: g = geometric probability distribution function

X ~ G ( p )

Read this as " X is a random variable with a geometric distribution ." The parameter is p ; p = the probability of a success for each trial.

Geometric formula

P ( X = x )= p 1 1 p x-1

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 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. X ~ G(0.02)

P ( X = 7)= 0.02 1 0.98 6 = 0.0177.

Using ti-83,83+,84,84+ calculator

To find the probability that X = 7,

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

To find the probability that X ≤ 7, 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 X ~ G(0.02) is:

This graph shows a geometric probability distribution. It consists of bars that peak at the left and slope downwards with each successive bar to the right. The values on the x-axis count the number of computer components tested until the defect is found. The y-axis is scaled from 0 to 0.02 in increments of 0.005.

The y -axis contains the probability of x , where X = 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, μ  = 50 .

The formula for the mean is μ = 1 p = 1 0.02 = 50

The formula for the variance is σ 2 = ( 1 p ) ( 1 p − 1 ) = ( 1 0.02 ) ( 1 0.02 − 1 ) = 2,450

The standard deviation is σ = ( 1 p ) ( 1 p − 1 ) = ( 1 0. 02 ) ( 1 0. 02 − 1 ) = 49.5

Try it

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 the TI-83+ or TI-84 calculator to find the answer.

P ( X = 9) = 0.0092

The lifetime risk of developing pancreatic cancer is about one in 78 (1.28%). Let X = 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: X ~ G ( 1 78 ) or X ~ G (0.0128).

  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 .
  1. P ( X = 10) = geometpdf(0.0128, 10) = 0.0114
  2. P ( X = 20) = geometpdf(0.0128, 20) = 0.01
    1. Mean = μ = 1 p = 1 0.0128 = 78
    2. Standard Deviation = σ  =  1 − p p 2 = 1 − 0.0128 0.0128 2 ≈ 77.6234
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Source:  OpenStax, Introduction to statistics i - stat 213 - university of calgary - ver2015revb. OpenStax CNX. Oct 21, 2015 Download for free at http://legacy.cnx.org/content/col11874/1.3
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