TITLE: Conditional probabilities, urns
QUESTION [5 upvotes]: I found this question interesting and apparently it has to do with conditional probabilities:
An urn contains six black balls and some white ones. Two balls are drawn simutaneously. They have the same color with probability 0.5. How many with balls are in
the urn?
As far as I am concerned I would say it is two white balls...

REPLY [2 votes]: Let there be $6$ black and $w$ white balls. The probability $P$ that we draw a pair of equally colored balls is given by
$$P={{6\choose 2}+{w\choose2}\over{6+w\choose2}}={6\cdot 5+w(w-1)\over(6+w)(5+w)}\ .$$
The condition $P={1\over2}$ leads to the quadratic equation $w^2-13w+30=0$ with the two solutions $w=3$ and $w=10$.