Problem: Plain old Monty Hall
(a) Make sure you understand why it is always better to switch in the Monty Hall problem described in class. Think about the problem from different angles. Keep it active in your mind for some time.
(b) Here's a convincing argument in favor of switching. We can simulate the game.
1. c ← 0 (this counts the number of wins)
2. n ← 0 (this counts the number of trials)
3. place the prize uniformly at random in one of the boxes, call this box x
4. choose a box uniformly at random, call this box y
5. decide that you will not switch no matter what (the simulation of the remainder of the game becomes irrelevant)
6. if x = y then c ← c + 1
7. n ← n + 1
8. print c/n
9. goto 3
It should be obvious that c/n will converge to 1/3 (why?), which is the prob- ability of winning if we adopt a no switch strategy. The simulation works the same way regardless of the actions taken by the host. Did we just prove that there is always a benefit in switching? (something to think about)
Assume that Monty Hall does not know where the prize is. He will simply open one of the remaining boxes at random with equal probability. Consider the case where an empty box is revealed. Show that there is no benefit in switching.
(d) How can we resolve the inconsistency between (b) and (c)?