What is the probability of your winning all of the money?

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August 16, 2019
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August 16, 2019

What is the probability of your winning all of the money?

You are playing rounds of a certain game against an opponent until one of you has won all of the other one’s betting money. At the start of each round, each of you stakes one dollar. The probability of winning any given round is equal to p, and the winner of a round gets the other player’s dollar. Your starting capital is a dollars, and your opponent’s starting capital is equal to b dollars. What is the probability of your winning all of the money? The renowned gambler’s formula is with p ≠ ½ (otherwise your probability of winning is equal to a/(a + b)). In order to prove this formula, argue first the recursion relation in which Pk is defined as the probability of your eventually winning all of the money, when your capital is k dollars and your opponent’s capital is a + b − k dollars (P0 = 0 and Pa+b = 1). Next, verify through substitution that the above formula is correct.

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