Question

You are a contestant on a game show. There are three doors, labeled A, B, and C. Behind two of the doors are prizes worth nothing, and behind the third door is a prize worth $10,000. The game show host knows which door contains the valued prize, but you don’t. You get first move, at which time you must choose one of the three doors. The game show host gets the second move, at which time he must open one of the doors that you didn’t select, revealing the contents behind that door and leaving the other two doors unopened. You get the third move, at which time you can choose to stay with your initial choice or switch your choice to the remaining unopened door. After you make your final choice, the door you selected in move 3 is opened, and you win the prize behind that door. Your preferences are to maximize your expected dollar payoff whereas the preference of the game show host is to keep the suspense going as long as possible. Under what circumstances, if ever, should you switch doors after the game show host opens a door?

Answer 1

if the contestant chooses A, the chance of him choosing the correct door is 1/3.

now, the chance of the prize being behind either of the two remaining doors is 2/3. i.e, the bracket {B, C} = 2/3

as the host has to keep the suspense he opens the door without the prize, let it be B.

now, as the contestant knows that he B does not have the prize and his chance of winning with A is 1/3 so he should always choose C as the bracket still has prob of 2/3 of containing the prize and the bracket no more has B.