28.2 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?
3The 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 asx = number of trials until the first success is observed (including the success trial)The probability distribution of x is called the geometric probability distribution.
7ExampleOver 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.
8Example - solutionThis 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
11A 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.
12A 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 = 0.21Find the mean = 1/p = 1/.7 1.43Find the standard deviation
14Example 8.48 p.550The 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?