3. Introduction We have been looking at Binomial Distributions: A family has 3 children. What is the probability they have 2 boys? A family has 3 children. What is the probability they have 2 boys? Johnny rolls a die 6 times. What is the probability that he will roll a 2 three times? Johnny rolls a die 6 times. What is the probability that he will roll a 2 three times? Now let’s look at these as Geometric Distributions: A family will have kids until they have a boy A family will have kids until they have a boy Johnny rolls a die until he rolls a 2 Johnny rolls a die until he rolls a 2

4. What Gives Us a Geometric Setting? There are 4 key elements necessary to have a geometric setting. They are as follows: Each observation is either a “Success” or “Failure” Each observation is either a “Success” or “Failure” The n observations are all independent. The n observations are all independent. The probability of success, p, is the same for each observation The probability of success, p, is the same for each observation ***The variable of interest is the number of trials required to obtain the first success ***The variable of interest is the number of trials required to obtain the first success Let’s look at these elements in the context of a scenario.

5. What Gives Us a Geometric Setting? Scientists have randomly selected 10 rats known to have the flu and will inoculate them with a vaccine until one is cured. Each observation is either a “Success” or “Failure” Each observation is either a “Success” or “Failure” In this case, the rats will either be Cured or Not Cured The variable of interest is the number of trials required to obtain the first success The variable of interest is the number of trials required to obtain the first success We are looking to find probabilities that the 1 st or the 2 nd or the 3 rd rat (etc.) gets cured The n observations are all independent. The n observations are all independent. The results of the vaccine in one rat do not effect the results for the next rat The results of the vaccine in one rat do not effect the results for the next rat The probability of success, p, is the same for each observation The probability of success, p, is the same for each observation The probability of success for each rat is.65

6. Examining the “Lingo” Geometric probabilities Probability of X=n is the probability that the 1 st success, X, is on the n th trial Probability of X=n is the probability that the 1 st success, X, is on the n th trial n = number of trials p = Probability of Success on any ONE observation *Remember you must 1 st have a geometric setting (meet the 4 criteria) before you can calculate a geometric probability Geometric Probability Distributions are technically infinite but as you increase trials, the probability will get closer and closer to 0

7. Finding Geometric Probabilities Below is the formula for calculating a geometric probability. To display a distribution, you just continue to calculate the probabilities Let’s look at the formula This is basically “failure*failure*failure…*success”. Where you have n-1 failures before you succeed Want to do this with a calculator function: geometpdf(p,x) where x is the trial you are looking for and p is the probability of success

8. Let’s Practice Using the Formula and the Calculator Allen Iverson is a career 73% free throw shooter. Find the probabilities below. 1. Find the probability that Allen makes his first shot. 2. Find the probability that it takes Allen two shots to make one. 3. Find the probability that Allen shoots five shots before he makes one. 4. Find the probability that Allen makes takes no more than four shots to make one. geometpdf(.73,1) =.73 geometpdf(.73,2) =.1971 geometpdf(.73,6) =.0010 geometpdf(.73,1) + geometpdf(.73,2) + geometpdf(.73,3) + geometpdf(.73,4)

9. Cumulative Function We can use the cumulative function for geometrics as well… They give you the probability of n trials or less They give you the probability of n trials or less geometcdf(p,x) geometcdf(p,x) This gives you the sum of the probabilities from 1 to X This gives you the sum of the probabilities from 1 to X Find the probability that Allen makes takes no more than four shots to make one. geometcdf(.73,4) =.9947 You do MORE THAN probabilities by subtracting the cdf from 1, just like in the binomial setting.

10. Get Some Answers!!! Find the probabilities below: Probability Allen takes more than 4 shots to make his first. Probability Allen takes more than 4 shots to make his first. Probability it takes Allen more than 6 shots to make his first. Probability it takes Allen more than 6 shots to make his first. Probability Allen makes his first shot in on his 8 th try. Probability Allen makes his first shot in on his 8 th try. 1 – geometcdf(.73,4) =.0053 1 – geometcdf(.73,6) =.0004 geometpdf(.73,8) =.00008

11. The Pesky Formula Below is the “by hand” formula used for MORE than probabilities Roll a die until a 3 is observed. The probability that it takes more than 6 rolls to observe a 3 is: P(X>6) = (1-1/6) 6 = (5/6) 6 =.335

12. Geometric Mean & Spread Geometric Mean = Geometric Variance = Geometric Standard Deviation =

13. Homework Read Pages 472,73 on Simulating Geometric Experiments Do Problem #’s 41 - 51