2008 Mock ARML 1 Problems/Problem 4
Problem
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.
Solution
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 .
See also
2008 Mock ARML 1 (Problems, Source) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 |