Source

I got this little problem from Rustan Leino, who got it from Leonardo de Moura.

I solved it and wrote up my solution.

Problem

There are three boxes, one containing two black balls, another containing two white balls, and another containing one black and one white ball. You select a ball uniformly randomly from a box selected uniformly randomly and you find you selected a white ball. What is the probability that the other ball in the same box is also white?

Solution
Reveal

The answer is $2/3,$ for the following reason.

We're looking for $P(\mbox{other ball is white} \mid \mbox{selected ball is white})$. This is $P(\mbox{other ball is white and selected ball is white})/P(\mbox{selected ball is white}).$ The numerator here is $1/3$ since you selected the boxes uniformly randomly and both balls will be white if and only if you selected the box containing two white balls. The denominator here is $1/2$ since all balls were equally likely to be picked and half of them are white. So the answer is $\frac{1/3}{1/2} = 2/3.$