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Question

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A and B throw a pair of dice alternately, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first.

Solution

Probabilities space left parenthesis Win right parenthesis equals space 3 over 36 space equals 1 over 12 Probabilities space left parenthesis lose right parenthesis space equals space 33 over 36 space equals space 11 over 12 straight P left parenthesis straight A space wins right parenthesis space equals space 1 over 12 plus 11 over 12 space straight x 11 over 12 space straight x 1 over 12 plus open parentheses 11 over 12 close parentheses to the power of 4 space straight x 1 over 12 plus........ straight a space equals 1 over 12 space straight r space equals 121 over 144 By space using space the space formula space of space inifinite space straight G. straight P. straight P left parenthesis straight A space wins right parenthesis space equals fraction numerator begin display style 1 over 12 end style over denominator 1 minus begin display style 121 over 144 end style end fraction space space equals space 12 over 23

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