The monty hall problem Afra Zomorodian January 20, 1998 Introduction This is a short report about the infamous"Monty Hall Problem. The report contains two solutions to the problem: an analytic and a numerical one. The analytic solution will use probability theory and corresponds to a mathematician's point of view in solving problems. The numerical solution simulates the problem on a large scale to arrive at the solution and therefore corresponds to computer scientist's point of view The monty hall Problem The Monty Hall Problem's origin is from the TV show, "Lets Make A Deal"hosted by Monty Hall. The statement of the problem is as follows [LR94 You are a contestant in a game show in which a prize is hidden behind one of three curtains you will win the prize if you select the correct curtain. After you have picked one curtain but before the curtain is lifted, the emcee lifts one of the other curtains, revealing an empty stage, and asks if you would like to switch from your current selection to the remaining curtain. How will your chances change if you switch? The question was originally proposed by a reader of"Ask Marilyn, a column by Marilyn vos Savant in Parade Magazine in 1990 and her solution caused an uproar among Mathematicians, as the answer to the problem is unintuitive: while most people would respond that switching should not matter, the contestant's chances for winning in fact double if she switches curtains. Part of the controversy, however, was caused by the lack of agreement on the statement of the problem itself We will use the above version. For accounts of the controversy as well as solutions and interactive applets, see Don98 Proof by Probability Theory without loss of generality, let us call the curtain picked by the contestant curtain a, the curtain opened by Monty Hall curtain b, and the third curtain c. We will define the following events A, B, and C are the events that the prize is behind curtains a, b, and c respectively .O is the event that Monty Hall opens curtain b The Monty Hall Problem can be restated as follows: Is Pr()= Prclo?The Monty Hall Problem Afra Zomorodian January 20, 1998 Introduction This is a short report about the infamous “Monty Hall Problem.” The report contains two solutions to the problem: an analytic and a numerical one. The analytic solution will use probability theory and corresponds to a mathematician’s point of view in solving problems. The numerical solution simulates the problem on a large scale to arrive at the solution and therefore corresponds to a computer scientist’s point of view. The Monty Hall Problem The Monty Hall Problem’s origin is from the TV show, “Let’s Make A Deal” hosted by Monty Hall. The statement of the problem is as follows [LR94]: “You are a contestant in a game show in which a prize is hidden behind one of three curtains. you will win the prize if you select the correct curtain. After you have picked one curtain but before the curtain is lifted, the emcee lifts one of the other curtains, revealing an empty stage, and asks if you would like to switch from your current selection to the remaining curtain. How will your chances change if you switch?” The question was originally proposed by a reader of “Ask Marilyn”, a column by Marilyn vos Savant in Parade Magazine in 1990 and her solution caused an uproar among Mathematicians, as the answer to the problem is unintuitive: while most people would respond that switching should not matter, the contestant’s chances for winning in fact double if she switches curtains. Part of the controversy, however, was caused by the lack of agreement on the statement of the problem itself. We will use the above version. For accounts of the controversy as well as solutions and interactive applets, see [Don98]. Proof by Probability Theory Without loss of generality, let us call the curtain picked by the contestant curtain a, the curtain opened by Monty Hall curtain b, and the third curtain c. We will define the following events: • A, B, and C are the events that the prize is behind curtains a, b, and c respectively. • O is the event that Monty Hall opens curtain b. The Monty Hall Problem can be restated as follows: Is P r{A | O} = P r{C | O}? 1