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  • 2007年07月06日

    “三扇门”逻辑题

    分类:

    19909月美国《广场杂志》(Monty Hall)“请教玛丽琳专栏,曾刊登如下逻辑题:
    假设你在进行一个游戏节目。现给三扇门供你选择:一扇门后面是一辆轿车,另两扇门后面分别都是一头山羊。你的目的当然是要想得到比较值钱的轿车,但你却并不能看到门后面的真实情况。主持人先让你作第一次选择。在你选择了一扇门后,知道其余两扇门后面是什么的主持人,打开了另一扇门给你看,而且,当然,那里有一头山羊。现在主持人告诉你,你还有一次选择的机会。那么,请你考虑一下,你是坚持第一次的选择不变,还是改变第一次的选择,更有可能得到轿车?

    正确答案:

    如果改变选择,你赢的概率有2/3当你第一次的选择是错的。

    如果不改变选择,你赢的概率只有1/3当你第一次的选择是正确的。

    原因:

      1.    It seems to make sense that you have a 1/3 chance of picking the correct door. This means, however, that since the probabilities must add up to one - and the car has to be somewhere - you also have a 2/3 chance of not picking the correct door. In other words, you are more likely not to win the car than to win it.

    Imagine that Monty opens a door and shows that there's only a goat behind it. Consider that the car is more likely to be behind a door other than the one you choose. Monty has just shown that one of those two doors - which together have the greater probability of concealing the car - actually conceals a goat. This means that you should definitely switch doors, because the remaining door now has a 2/3 chance of concealing the car. Why? Well, your first choice still has a 1/3 probability of being the correct door, so the additional 2/3 probability must be somewhere else. Since you know that one of the two doors that previously shared the 2/3 probability does not hide the car, you should switch to the other door, which still has a 2/3 chance of concealing the car.

      2.    What if there were 1,000 doors? You would have a 1/1,000 chance of picking the correct door. If Monty opens 998 doors, all of them with goats behind them, the door that you chose first will still have a 1/1,000 chance of being the one that conceals the car, but the other remaining door will have a 999/1,000 probability of being the door that is concealing the car. Here switching sounds like a pretty good idea.

     

     参考文章:

      1.      The Monty Hall Problem

      2.      Solution to Puzzle No. 4 - three doors

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