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 |