TITLE: Is there a quick way to justify that this elementary probability is equal to $\frac23$?
QUESTION [8 upvotes]: I just solved this problem with the conditional probability formula and after a while the answer was surprisingly $\frac23$.
I believe there must be a tricky short way to calculate it.
Can somebody help me?

There are $n$ urns of which the $r$th contains $r-1$ red balls and $n-r$ magenta balls.  You pick an urn at random and remove two balls at random without replacement.  Find the probability that: the second ball is magenta, given that the first is magenta.

REPLY [4 votes]: Imagine a row of $n$ urns with a single uncolored ball in each of them. One of the urns is selected at random, and its ball is colored white. The balls to the left of the white ball are colored red, and the balls to the right of the white ball are colored magenta. Now two more urns are selected at random, and their contents inspected. With probability ${1\over3}$ the white ball is the middle of the three, hence with probability ${2\over3}$ the contents of the other two urns are of the same color.