2008 Mock ARML 1 Problems/Problem 4
There are black balls and white ball in a hat. A turn consists of picking a ball from the hat and replacing it with one of the opposite color. Compute the probability that, after a sequence of turns, there are black balls in the hat before there are white balls.
Let denote the probability of reaching black balls before white balls from a position of black balls and white balls. The probability that we have black balls after a turn is (note that ), and that we have black balls is . Similarly, if we start with black balls, after a turn there is a probability of ending with black balls and a probability of ending with balls. Thus, we have the recursions By symmetry, ; substitution into yields . Now, and form a two equation linear system which can be solved to find that .
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