3rd Australian 1982 Problem A1 If you toss a fair coin n+1 times and I toss it n times, what is the probability that you get more heads?
1/2 Solution Suppose the prob of us getting equal numbers of heads in n tosses each is p. Then by symmetry the prob you get more is (1-p)/2. Now your last toss cannot worsen your position. If you were ahead at n, you will still be ahead whatever the outcome. If we were level at n, then you have 1/2 chance of moving ahead. So your chance of winning is 1/2 - p/2 + p/2 = 1/2. Comment. Of course, you can also slog out the relevant probabilities. Solutions are also available in Australian Mathematical Olympiads 1979-1995 by H Lausch and P J Taylor, ISBN 0858896451, published by Australian Mathematics Trust, can be ordered by email. 3rd Aus 1982 © John Scholes jscholes@kalva.demon.co.uk 2 January 2004 Last updated/corrected 2 Jan 04