26th IMO 1985 shortlist Problem 13 A die is tossed repeatedly. A wins if it is 1 or 2 on two consecutive tosses. B wins if it is 3 - 6 on two consecutive tosses. Find the probability of each player winning if the die is tossed at most 5 times. Find the probability of each player winning if the die is tossed until a player wins. Solution
: 5 throws or less: A 55/243, B 176/243; arbitrarily many throws: A 5/21, B 16/21. Solution by Demetres Christofides Denote an outcome of 1 or 2 as L and an outcome of 3, 4, 5, or 6 as H. For 5 tosses or less we have: LL A wins prob. 27/243 LHLL A wins prob 6/243 LHLHL draw prob 4/243 LHLHH B wins prob 8/243 LHH B wins prob 36/243 HLL A wins prob 18/243 HLHLL A wins prob 4/243 HLHLH draw prob 8/243 HLHH B wins prob 24/243 HH B wins prob 108/243 Summarising, the prob of A winning is 55/243, of B winning is 176/243 and of a draw is 12/243. With an arbitrary number of throws A wins with LL, LHLL, LHLHLL, ... or with HLL, HLHLL, HLHLHLL, ... . Probability (1 + 2/3) 1/9 (1 + 2/9 + (2/9) 2 + ... ) = (5/3)(1/7) = 5/21. But no draw is possible, so B wins with prob 16/21. 26th IMO shortlist 1985 © John Scholes jscholes@kalva.demon.co.uk 11 Sep 2002