You are one of 20 logicians worldwide to receive this letter. You don't know each other, but you all think alike. An address is given, and you are told that if a letter is received from one and only one of you, all 20 will equally split a large sum of money. If no letters are received, or more than one, no prize.

What would you to for a chance of winning the prize? What could you do in order to maximise the probability of winning?

Assuming that you decide on the probabilistic approach, what is the optimum probability to use? In fact, with N logicians, it's easy to prove that 1/N is optimal.

If N people send a letter with independent probabilities of p, the odds of receiving exactly one letter are N * p * (1-p)^(N-1). To find the maximum of this, take the derivative with respect to p, set the result to zero and solve. The derivative is N*(1-p)*(N-1) - N*(N-1)*p*(1-p)^(N-2). This is zero when p = 1/N.

When N people send a letter with probability 1/N, the probability of exactly one letter being sent simplifies to ((N-1)/N)^(N-1). As N becomes large, this value approaches 1/e.

Posted by Jim Lyon on 2002-12-11 06:58:18