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Geometric Probabilities Presentation 5.6

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Lottery Tickets In our ignorance, we are going to invest in our future by purchasing one lottery ticket every week. We are smart enough to know that once we win we better stop playing. The chance of winning on the type of lottery tickets we purchase is 0.05 or 1 in 20. Check to verify that this is a geometric setting.

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Lottery Tickets There are only two outcomes. –Success (winning ticket) or failure (losing ticket) Keep going until 1 st success –We will stop buying tickets after we win Trials are independent –Winning or losing on any given ticket will not affect your chance of winning or losing on the next (or any other) ticket Probability of success is constant –Each ticket has the same chance (0.05) of being a winning ticket

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Lottery Tickets Some questions to consider: –What is the probability of winning on the first ticket? –What is the probability of winning on the second ticket? –What is the probability of winning on the third ticket? –What is the probability of winning on the fourth ticket?

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Lottery Tickets Some questions to consider: –What is the probability of winning on the first ticket? This would be P(X=1) meaning the probability of getting success on the 1 st try and = –What is the probability of winning on the second ticket? This would be P(X=2) meaning the probability of getting success on the 2 nd try and = (0.95)(0.05) = –Think you have to fail first, then succeed. –What is the probability of winning on the third ticket? This would be P(X=3) meaning the probability of getting success on the 3 rd try and = (0.95)(0.95)(0.05) = –Think you have to fail, fail, then succeed. –What is the probability of winning on the fourth ticket? This would be P(X=4) meaning the probability of getting success on the 2 nd try and = (0.95)(0.95)(0.95)(0.05) = –Think you have to fail, fail, fail, then succeed.

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Lottery Tickets Some questions to consider: –What is the probability of winning on the first ticket? This can also be done by using the calculator and geometpdf. Geometpdf is found under distribution Geometpdf (p, x) where p is the probability of success and x is the trial on which you get success Geometpdf(.05,1) =.05 –What is the probability of winning on the second ticket? Geometpdf(.05,2) =.0475 –What is the probability of winning on the third ticket? Geometpdf(.05,3) =.0451 –What is the probability of winning on the fourth ticket? Geometpdf(.05,4) =.0429

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Lottery Tickets Some other questions to consider: –What is the probability of winning by the tenth ticket? You can also use geometcdf (p,x) which will find the probability of success by the x trial. Geometcdf(.05,10) =.4013 This is the chance that you will get success by the tenth ticket –How many tickets would you expect to have to buy? Since the probability is.05 or 1/20, you would, on the average, expect to buy 20 tickets. This is the geometric mean which is 1/p. –In this case 1/0.05 = 20

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Lottery Tickets Some other questions to consider: –What is the probability of winning only after the 52 nd ticket? You can also use geometcdf (p,x) which will find the probability of success by the x trial. Geometcdf(.05,52) =.9306 This is the chance that you will get success by the 52 nd ticket So, take the complement or =.0694

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Geometric Distributions Similar to binomial except you go until your first success With binomial it is possible to have X=0 –Indicating 0 successes in n trials With geometric, it is NOT possible for X=0 –Since you go until first success you must have at least 1 trial. Be able to conduct simulations with geometric settings

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Geometric Probabilities This concludes this presentation.

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