Date: Wed, 12 May 2000 22:24:57 -0400

From: Alex Bogomolny

Dear Clifton:

I suspect that both your answers are wrong. The first one one is similar to the famous Achilles and Tortoise paradox. The fact that every try has a finite probability does not guaranty a win even after an infinite number of steps. Your second solution shows why. The total probability of a win after an infinite number of steps is the sum of a series which might have happpened to be less than 1.

Your second answer has a numerical flaw. Assume that
the probability of a win on the n^{th} try is p_{n} and let
q_{n} = 1 - p_{n}. (p_{n} = 2^{-n}, but this is not important.)

We can write

1 =

p_{1} + q_{1} =

p_{1} + q_{1}(p_{2} + q_{2}) =

p_{1} + q_{1}p_{2} + q_{1}q_{2}(p_{3} + q_{3}) =

p_{1} + q_{1}p_{2} + q_{1}q_{2}p_{3} + q_{1}q_{2}q_{3}(p_{4} + q_{4}) =

p_{1} + q_{1}p_{2} + q_{1}q_{2}p_{3} + q_{1}q_{2}q_{3}p_{4} + q_{1}q_{2}q_{3}q_{4}p_{5} + ...

The sum of the first n terms is the probability of a win after n tries. The sum of the series is naturally 1! Try to find an error in your calculations.

The answer to your question is thus that the probability of a win after an infinite number of steps is 1. However, note that this does not guarantee you a win. Events that occur with probability 1 are not bound to happen.

For example, probability that a random number on the unit interval is irrational is 1. However, when picking a random number from that interval, one can get a rational number, although probability of this event is 0.

All the best,

Alexander Bogomolny