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In a 1970’s game show, a contestant was given a choice of three doors of which one contained a prize. The other two doors contained gag gifts like a chicken or a donkey. After the contestant
chose an initial door, the host of the show then revealed an empty door among the two un-chosen doors, and asks the contestant if he or she would like to switch to the other un-chosen door. The
question is should the contestant switch. Does the probability of winning increase by switching to the remaining door?

  • Probability -

    Yes, the probability of winning improves from 1/3 to 2/3 by switching. See

  • Probability -


    Vos Savant's solution

    The solution presented by vos Savant in Parade (vos Savant 1990b) shows the three possible arrangements of one car and two goats behind three doors and the result of switching or staying after initially picking Door 1 in each case:
    Door 1 Door 2 Door 3 result if switching result if staying
    Car Goat Goat Goat Car
    Goat Car Goat Car Goat
    Goat Goat Car Car Goat

    A player who stays with the initial choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three. The probability of winning by staying with the initial choice is therefore 1/3, while the probability of winning by switching is 2/3.

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