Suppose you are a prisoner. The king gives you $100$ marbles, $50$ black and $50$ white. You have to put all the marbles into $2$ urns such that none of the urns is empty. The king will then choose an urn at random and from that a marble at random. If the marble is white, you will be set free, otherwise you will be sentenced to death. So you have to distribute the marbles in such a way so as to maximize the chance of survival.
Basically you have to maximize the probability that the king chooses white marble.
My attempt : Let there be $b$ black and $w$ white marbles in the 1st urn . Therefore, there will be $50-b$ black and $50-w$ white marbles in the 2nd urn.
Therefore P(Survival) = P(Choosing white marble) = $\frac{1}{2}\cdot\frac{w}{w+b} +\frac{1}{2}\cdot\frac{50-w}{100-w-b}$ .
Now I need to maximize the above equation but I don't know how.
Any ideas?
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