8.2 The Geometric Distribution 1.What is the geometric setting? 2.How do you calculate the probability of getting the first success on the n th trial?

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8.2 The Geometric Distribution 1.What is the geometric setting? 2.How do you calculate the probability of getting the first success on the n th trial? 3.How do you calculate the means and variance of a geometric distribution? 4.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: 1.The trials are independent. 2.Each trial can result in one of two possible outcomes, success and failure. 3.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) = (1 –  ) x-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  = Let x = number of customers at the drive- in window before a customer makes a deposit. The desired probability is

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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