Probability the game show problem

Transcription

1 the game show problem Dr. Maureen Tingley For today, Pr means probability.. is hard.. Probabilities are always between 0 and (inclusive).. Sometimes it makes intuitive sense to multiply probabilities. 4. Sometimes it makes intuitive sense to add probabilities. Tree diagrams can be used to answer complicated problems. The conditional probability of A, given B: Pr(A B) = Pr(events A and B both occur) Pr(event B occurs) Quotation from Let s Make a Deal: The Player s Dilemma, J. P. Morgan, N. R. Chaganty, R. C. Dahiya and M. J. Doviak. The American Statistician, November 99, p. 84: In a trio of columns, titled Ask Marilyn, in Parade Magazine (vos Savant 990, 99) the following question was posed: Suppose you re on a game show and given a of three doors. Behind one is a car; behind the others are goats. You pick door No., and the host, who knows what s behind them, opens No., which has a goat. He then asks if you want to pick No.. Should you switch? (vos Savant 990b). Marilyn vos Savant, the column author and reportedly holder of the world s highest I.Q., replied in the September article Yes you should switch....in the December article letters from three Ph.D.s appeared saying that vos Savant s answer was wrong....by the February article a full scale furor had erupted; vos Savant reported I m receiving thousands of letters, nearly all insisting I m wrong.... Of the letters from the general public, 9% are against my answer; and of letters from universities, 65% are against my answer. Nevertheless vos Savant does not back down.

2 The controversy arose because people were making different assumptions about the host s strategy, and about which probability should be calculated. Consider three different strategies for the host. In all cases, the game ends as soon as the car is revealed, or the contestant s door is opened.. Randomly open any door. (So the game could end at this point.). Randomly open one of the two unselected doors. (So the game could end at this point.). Randomly open an unselected door that has a goat. (So the game will continue.) No matter what strategy the host uses, there are four probabilities of interest. The first probability is an unconditional probability, calculated before the game starts, and called the a priori probability that the contestant will win. Pr(Contestant wins) The next three probabilities are conditional probabilities, calculated after the game has started. Pr(Contestant wins Host has revealed a goat, and the game is not yet over) Pr(Contestant wins Host has revealed a goat) Pr(Contestant wins Contestant chose door and Host has revealed a goat behind door ). On the following pages, A stands for automobile and G stands for goat. Work through these pages (one for each of the above host strategies) in teams of two: One team member should assume that, given the opportunity, the contestant will always switch. The other team member should assume that, given the opportunity, the contestant will never switch. Then come back and answer the big question: Should the contestant switch?

3 Host strategy : Randomly open any door. Pr(Contestant wins Contestant chose door and Host revealed a goat behind door ) =

4 Host strategy : Randomly open one of the two unselected doors. Pr(Contestant wins Contestant chose door and Host revealed a goat behind door ) = 4

5 Host strategy : Randomly open an unselected door that has a goat. Pr(Contestant wins Contestant chose door and Host revealed a goat behind door ) = 5