Question

126. An article entitled "Behind Monty Hall's Doors: Puzzle, De- bate and Answer?" appeared in the Sunday New York Times on July 21, 1991. The article discussed the debate that was raging among mathematicians, readers of the "Ask Marilyn" column of Parade Magazine and the fans of the TV game show "Let's Make a Deal." The argument began in Septem- ber, 1990 when Ms. Vos Savant, who is listed in the Guinness Book of World Records Hall of Fame for the highest IQ, posed the following question in her "Ask Marilyn" column of Parade Magazine: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?" Is it to your advantage to take the switch? Since Ms. Vos Savant gave her answer, she has received ap- proximately 10,000 letters with the great majority disagreeing with her answer. The most vebement from mathematicians and scientists who have called her the "goat." Her answer has been debated from the halls of the CLA to the mathemati- cians of MIT.Even Monty Hall conducted a simulation of the problem in his dining room. After which Monty came to the conclusion that Ms. Vos Savant's critics were dead wrong when the host is required to open a door all the time and offer you a switch. However, Monty suggested since he hasa choice on the show as to whether he allows the contestant the choice to switch or not, then the question cannot be answered without knowing the motivation of the host. As Monty Hall said: "If he (the host) has the choice whether to allow the switch or not, beware. Caveat emptor. It all depends on his mood. My only advice is, if you can get me to offer you $5,000 not to open the door, take the money and go homé." a) Monty Hall states in the article that the odds on the car being behind door No. 1 are still only 1 in 3 even after he opened another door to reveal a goat. Explain why this is true. b) Some people argued with Ms Vos Savant that if one door is shown to be a loser, that information changes the prob- ability of either remaining choice-neither of which has any reason to be more likely-to 12. Explain why this is not correct reasoning

Answer 1

a)

because

The chance that you originally picked the correct door always stays the same. It was 1 out of a 3 when you first picked it, and it is 1 out of a 3 when the last door is left. When doors are eliminated from the remaining ones, they do not reveal any new information about your past choice, only about future choice

b)

because sum of probability has to be 1

since

the probability of car in door which was chose earlier = 1/3   from a) part

probability of car in other door = 1-1/3