IMO shortlist 1985/13 solution

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

Answer: 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