Problem
of the
Week

This week's problem was submitted by Professor David Quigg of the Bradley Mathematics Department.  Thanks, Marathon Man!

PROBLEM 70

Two gamblers take turns rolling a fair N-sided die, with N at least 5.  The faces of the die are labeled with the numbers 1 to N.  The first player starts.  If he rolls an N or an N-1, he wins and the game is over.  Otherwise, the other player rolls the die; if she rolls a 1, 2, or 3, she wins and the game is over.  Play continues, with the players alternating rolls until one of them wins.

What is the probability that the first player will win?  Are there any values of N for which the first player has at least an even chance of winning?


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