The Three Doors Paradox

    Alf appears on a gameshow and is shown three doors. Behind one door, he is told, is a car. behind each of the other two doors is a goat. Alf is asked to choose a door and he does so. The gameshow host then opens one of the remaining two doors to reveal a goat. Alf is asked whether he wants stick with his originally chosen door or switch to the other unopened door. His prize is whatever lies behind his final choice of door.
    Assuming Alf wants to win the car, should he switch?

    The answer is that Alf should switch. If he sticks, he will win if his first choice hid the car (probability 1/3). If he switches, he will win if his first choice hid a goat (probability 2/3).

    This "paradox" is interesting because of the vehement insistance it induces in otherwise intelligent people that it makes no difference whether Alf swaps or not. The arguments used to support this position all tend to boil down to the erroneous premise "There are two doors so it must be 50-50" but folk will argue long and hard over this one.

    It can be extremely difficult to persuade such people of the validity of the 1/3 vs 2/3 argument. One approach is to postulate a Shub Nigguruthian variation of the game with 1000 doors and 999 goats. Alf picks a door and the host flings open 998 of the remaining 999 doors revealing goats. The 99.9% chance of the unpicked and unopened door hiding the car is sometimes enough to dent 50-50 convitions.

Newcomb's Paradox

            

Back to Ian Bell's home page