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
![\begin{align} f(4) &= \frac 15 \cdot [1] + \frac 45 \cdot f(3) \\ f(3) &= \frac 25\cdot f(4) + \frac 35 \cdot f(2) \end{align}](http://latex.artofproblemsolving.com/7/9/8/7989f1e9ff51a30d3e7856869eb5f85de943a8b0.png)
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 | ||
