From: hrubin@odds.stat.purdue.edu (Herman Rubin) Subject: Re: A Probability Paradox from David Gale Date: 25 Nov 2000 20:15:39 -0500 Newsgroups: sci.math Summary: [missing] In article <20001125190510.22898.00001651@ng-fq1.aol.com>, LMWapner wrote: >Hello: >The following antimony appears in David Gale's book, "Tracking the Automatic >Ant". It reminds me of the "Two Envelope Paradox", but is a bit more >disturbing. >"In a certain casino one can play the following game. The house posts a >positive integer n. In this game it is you the customer who are invited to >toss a fair coin until it falls tails. If you tossed n-1 times then you pay >the house 8^(n-1) dollars, but if you tossed n+1 times, you win 8^n dollars >from the house. In all other cases the payoff is zero. Since the probability >of tossing exactly n times is 2^(-n), your expect winnings are (8^n)/[2^(n+1)] >- [8^(n-1)]/2^(n-1) = 4^(n-1), for n>1 and 2 for n=1. So your expected gain, >which is the house's expected loss, is positive. >But now it turns out that the house arrived at the number n by tossing that >same fair coin and counting the number of tosses up to and including the first >tails. Thus, you and the house are behaving in a completely symmetric manner. >Each of you tosses the coin, and if the number of tosses happens to be the >consecutive integers n and n+1, then the n-tosser pays the (n+1)-tosser 8^n >dollars. But we've just seen that the game is to your advantage as measured by >expectation no matter what number the house announces. How can there be this >asymmetry in a completely symmetric game?" >I have yet to resolve this. Help? The problem is that the expected payoff does not exist. That E(X|Y) > Y and E(Y|X) > X is paradoxical, but not contradictory, as neither E(X) nor E(Y) is finite. A similar situation arises in analysis, when f(x,y) = -f(y,x), and the integral of f with respect to one variable is positive for every value of the other; in fact, the problems are essentially the same, as the probability problem uses an integral with respect to a different measure. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558