Two players A and B take turns rolling a single fair six-sided die,…
Question
Two players A and B take turns rolling a single fair six-sided die, with A rolling first and then alternating. A wins instantly if A ever rolls a 6 on A's turn; B wins instantly if B ever rolls a 1 on B's turn. If a player does not roll the winning face on a turn, play passes to the opponent. Play continues until someone wins. Let the probability that A wins be m/n in lowest terms. Find m+n.
Step-by-step solution
Let a = P(A ultimately wins | it is currently A's turn) and b = P(A ultimately wins | it is currently B's turn).
On A's turn: A rolls a 6 with prob 1/6 and wins (contributing 1); otherwise (prob 5/6) it becomes B's turn, contributing b. So
On B's turn: B rolls a 1 with prob 1/6 and B wins (so A's winning prob is 0); otherwise (prob 5/6) it becomes A's turn, contributing a. So
Final answer17
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