I place one dollar on the table. I repeatedly flip a coin. Each time the coin comes up heads, I double the money on the table. The first time the coin comes up tails, you take the money and the game is over.

What's a fair admission price for the game?

Would you play the game with me for $100?

Actually it is quite clear. It's just that, in expressing it as a game of chance, Jim confused you, because real casinos and gamblers only play games based on odds. Long - term statistics are only used to evaluate the house' edge (and even then it's clearly based on the odds.

Matematically, what Jim is asking is Can we determine an average length for a "streak" of heads. The answer to his question of a "fair price" would be 2^(s - 1) the same as the payout after that average streak. (I'm not sure, but I believe that because of the doubling (as opposed to simply adding a new dollar), by "average" here we are looking for an analog of the geometric mean

  n      n
(  √[   ∏i(n)])
        i=1

rather than an analog of the arithmetic mean

     n
([   ∑i(n)] / n)
    i=1

In any case, that is the real task.

Posted by TomM on 2002-10-09 18:05:21