Let's Make a Deal Puzzle
Find the optimal strategy and the expected value of an outcome in the following game.
There
are three urns. Two of them are empty and one has a 1 dollar coin in
it, but you don't know which one. You bet, at random, 1 dollar on
one of the three urns. After you placed your bet, the dealer picks up,
out of the remaining two urns, an empty one (at least one of them has
to be empty) and shows you that it is empty. Now, you may either change
the urn you are betting on, or you may stay with your original bet. If
you selected, eventually, an empty urn then you lost your bet (1
dollar). Otherwise, you won the contents of the urn you selected (1
dollar, that is). In the latter case, of course, you keep the dollar
you bet.
Hint. To be perfectly correct, you need to do all the following steps.
1. Find a suitable set of elementary events.
2. Find the probability distribution on that set.
3. Find a suitable random variable that measures the outcomes of the game.
4. Write a formula for the mentioned above expected value.
5. Compute that value for all possible strategies in that game (there should be no more than three).
5. Pick up the one that yields the biggest expected value.
7. The value corresponding to that strategy is the answer.
Note. An informal
solution is really easy, but the main point here is to use rigorous
calculations rather than an intuitive (or naive) method.