The Practice of Statistics Daniel S. Yates The Practice of Statistics Third Edition Chapter 8: The Binomial and Geometric Distributions 8.2 The Geometric Distribution Copyright © 2008 by W. H. Freeman & Company

8.2 The Geometric Distribution What is the geometric setting? How do you calculate the probability of getting the first success on the nth trial? How do you calculate the means and variance of a geometric distribution? How do you calculate the probability that it takes more than n trials to see the first success for a geometric random variable?

The Geometric Distribution Suppose an experiment consists of a sequence of trials with the following conditions: The trials are independent. Each trial can result in one of two possible outcomes, success and failure. The probability of success is the same for all trials. A geometric random variable is defined as x = number of trials until the first success is observed (including the success trial) The probability distribution of x is called the geometric probability distribution.

The Geometric Distribution If x is a geometric random variable with probability of success =  for each trial, then p(x = n) = (1 – )n-1  x = 1, 2, 3, …

Example Over a very long period of time, it has been noted that on Friday’s 25% of the customers at the drive-in window at the bank make deposits. What is the probability that it takes 4 customers at the drive-in window before the first one makes a deposit.

Example - solution This problem is a geometric distribution problem with  = 0.25. Let x = number of customers at the drive-in window before a customer makes a deposit. The desired probability is

Geometric Probability

A sharpshooter normally hits the target 70% of the time. Find the probability that her first hit is on the second shot. Find the mean and the standard deviation of this geometric distribution.

A sharpshooter normally hits the target 70% of the time. Find the probability that her first hit is on the second shot. P(2)=p(1-p) n-1 = .7(.3)2-1 = 0.21 Find the mean  = 1/p = 1/.7 1.43 Find the standard deviation

Example 8.48 p.550 The State Department is trying to identify an individual who speaks Farsi to fill a foreign embassy position. They have determined that 4% of the applicant pool are fluent in Farsi. a). If applicants are contacted randomly, how many individuals can they expect to interview in order to find one who is fluent in Farsi? b). What is the probability that they will have to interview more than 25 until they find one who speaks Farsi?