Three People Toss a Coin

Date: 03/22/2002 at 03:36:27
From: George Corston
Subject: Probability

Three people (A, B, C) each toss a coin in order. The winner is the 
first to get a 'tail'. Expressed as fractions, what are their 
respective chances of winning?

Date: 03/22/2002 at 07:05:13
From: Doctor Anthony
Subject: Re: Probability

If a = the probability that A wins, then he can win with probability 
1/2 or get back to a with probability 1/8 (i.e. A, B and C all fail on 
their first turns).

    a = 1/2 + (1/8)a

   (7/8)a = 1/2

        a = 8/14 = 4/7

Similarly if b = the probability that B wins, he can win on the first 
turn with probability 1/4 (i.e. A loses, B wins) or he can get back to 
b with probability 1/8.

     b = 1/4 + (1/8)b

    (7/8)b = 1/4

         b = 2/7

and finally  c = 1/7

So the probabilities are, respectively,  4/7, 2/7, and 1/7.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/