Lecture 4.2 (cont.) Geometric Random VariablesBinomial ExperimentsThe Geometric ModelThe Geometric Model (cont.)Slide 5ExampleExample (cont.)Slide 8Slide 9Slide 10Slide 11Question from first slide1Lecture 4.2 (cont.)Geometric Random VariablesGeometric Probability DistributionsThrough 2/24/2011 NC State’s free-throw percentage was 69.6 (146th of 345 in Div. 1). In the 2/26/2011 game with GaTech what was the probability that the first missed free-throw by the ‘Pack occurs on the 5th attempt?2Binomial Experimentsn identical trialsn specified in advance2 outcomes on each trialusually referred to as “success” and “failure”p “success” probability; q=1-p “failure” probability; remain constant from trial to trialtrials are independentThe binomial rv counts the number of successes in the n trials3The Geometric ModelA geometric random variable counts the number of trials until the first success is observed.A geometric random variable is completely specified by one parameter, p, the probability of success, and is denoted Geom(p).Unlike a binomial random variable, the number of trials is not fixed4The Geometric Model (cont.)Geometric probability model for Bernoulli trials: Geom(p)p = probability of successq = 1 – p = probability of failureX = # of trials until the first success occursp(x) = P(X = x) = qx-1p, x = 1, 2, 3, 4,…1( )E Xpm= =2qps =5The Geometric Model (cont.)The 10% condition: the trials must be independent. If that assumption is violated, it is still okay to proceed as long as the sample is smaller than 10% of the population.Example: 3% of 33,000 NCSU students are from New Jersey. If NCSU students are selected 1 at a time, what is the probability that the first student from New Jersey is the 15th student selected?6ExampleThe American Red Cross says that about 11% of the U.S. population has Type B blood. A blood drive is being held in your area.1. How many blood donors should the American Red Cross expect to collect from until it gets the first donor with Type B blood?Success=donor has Type B bloodX=number of donors until get first donor with Type B blood1 1.11; ( ) 9.09.11p E Xp= = = =7Example (cont.)The American Red Cross says that about 11% of the U.S. population has Type B blood. A blood drive is being held in your area.2. What is the probability that the fourth blood donor is the first donor with Type B blood?4 1 4 1 3(4) (.89) (.11) .89 .11 .0775p q p- -= � = = � =8Example (cont.)The American Red Cross says that about 11% of the U.S. population has Type B blood. A blood drive is being held in your area.3. What is the probability that the first Type B blood donor is among the first four people in line?0 1 2 3.11;have to find(1) (2) (3) (4)(.89 .11) (.89 .11) (.89 .11) (.89 .11).11 .0979 .087 .078 .3729pp p p p=+ + += � + � + � + �= + + + =90 21 3(1) .9 .1 .1 (3) .9 .1 .081(2) .9 .1 .09 (4) .9 .1 .07291 1( ) 10.1p pp pE Xp= � = = � == � = = � == = =100 21 3(1) .75 .25 .25 (3) .75 .25 .141(2) .75 .25 .1875 (4) .75 .25 .10551 1( ) 4.25p pp pE Xp= � = = � == � = = � == = =11ExampleShanille O’Keal is a WNBA player who makes 25% of her 3-point attempts.1. The expected number of attempts until she makes her first 3-point shot is what value?2. What is the probability that the first 3-point shot she makes occurs on her 3rd attempt?2(3) .75 .25 .141p = � =1 1( ) 4.25E Xp= = =Question from first slide Through 2/24/2011 NC State’s free-throw percentage was 69.6%. In the game with GaTech what was the probability that the first missed free-throw by the ‘Pack occurs on the 5th attempt?“Success” = missed free throwSuccess p = 1 - .696 = .304p(5) = .6964  .304 =