Qua locus

Keeping both eyes on the long game.

A cute little question about coins


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Jack is tossing a fair coin. John believes in gambler’s luck, i.e. that if Jack has tossed more heads than tails so far then the next toss is more likely to be a tail. This way, in the long run things balance out. Jim knows that previous coin tosses have no effect on the next toss. John and Jim start betting with each other on the result of each coin toss. Who is likely to win more of the bets, John or Jim?

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Written by Christo Fogelberg

August 26, 02013 at 21:52

Posted in Puzzles

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6 Responses

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  1. Are they each betting the same amount on each toss?

    qwandor

    September 4, 02013 at 21:28

    • Unless they change their beliefs about gamblers luck I don’t think it will make any difference

      Christo Fogelberg

      September 5, 02013 at 22:35

      • Well, if they can change the amounts they are betting, then Jack can exploit John’s incorrect model to make a small profit.

        For example, suppose Jack happens to toss 4 heads in a row. Jim could then offer to bet $0.90 to John’s $1 that the next toss will also be a head. John believes a tail is much more likely so takes the bet. However, as each is equally likely, we would expect Jim to make a small profit on average.

        qwandor

        September 5, 02013 at 22:53

      • You’re right – silly of me to miss that!

        Christo Fogelberg

        September 9, 02013 at 21:48

      • On the other hand, the question you actually asked was who is likely to win more bets rather than who is likely to make money, so perhaps I am misunderstanding. If we are only interested in the number of bets won, then I would expect them both to win the same number in the long term, as their beliefs do not affect the behaviour of the coin.

        qwandor

        September 10, 02013 at 12:36

      • Yeah, that’s it. Intuitively, it would seem that having knowledge about another person’s bias should enable you to outguess them (and you could with adaptive betting, like you point out). It’s really interesting to me that sometimes a bias about randomness cannot be exploited in the same way.

        Christo Fogelberg

        September 11, 02013 at 21:07


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