On a game show, there are three doors, behind one of which is a prize. I choose a door and the host opens one of the other doors that has no prize behind it. I get to switch my door choice if I wish.
Now suppose we have three positive numbers $p_1$, $p_2$, $p_3$ such that $p_1+p_2+p_3=1$ and the prize is behind door $i$ with probability $p_i$. By labeling the doors suitably we can assume $p_1>p_2>p_3$. Assume that you know the probabilities $p_1$, $p_2$, $p_3$ associated to each door. What is the strategy that maximizes my chances of winning the prize?