Several posts ago I posed the famous Monty Hall three-door problem. Monty has placed $60,000 behind one of three closed doors on the stage. Behind the other two are goats. You choose a door -- say, Door 1. Monty then has his assistant open Door 3 to reveal a goat. He then offers to let you keep Door 1, to let you switch to Door 2, or to accept $35,000 (on the assumption that you are risk-neutral). The question, recall, is which of those three options you, the contestant, should choose.
You should switch to Door 2. (More generally, you should switch to the unopened door that you did not originally pick.) Trying to explain this convincingly is, however, a challenge. I'm going to accept that challenge in the next several posts.
Here's the gist of the matter. If you switch, you will win two-thirds of the time. When you chose Door 1, the probability that the cash prize lies behind any one of the three doors is one-third. The probability that it lies behind either Door 2 or Door 3 is two-thirds. Once Monty has shown you that there is a goat behind Door 3, all two-thirds of the probability that the prize lies behind Door 2 or Door 3 falls to Door 2. (There's a crucial assumption that I will get to in a moment.) So, the expected value of Door 1 is $20,000 (1/3 x $60,000), and the expected value of Door 2 (after Monty has Door 3 opened) is $40,000 (2/3 x $60,000). Therefore, the expected value of Door 2 is greater than that of either Door 1 or the $35,000 cash. Switch.
Of course, no one -- on first hearing -- is convinced by that line of argument. Most people think that the one-third probability that was initially with Door 3 but has now been revealed to be zero shifts in equal amounts to Doors 1 and 2. The argument made is, "Each of the still-closed doors gets half of the one-third that used to lie with Door 3, so that 1/6 of the probability formerly with Door 3 goes to Door 1 and 1/6 to Door 2. So, each of those doors now has a 1/2 chance of hiding the $60,000. So, each of the closed doors has an expected value of $30,000. As a result, I'll take the cash." That's incorrect.
I said above that there is a crucial assumption that bears on all this. Here it is: Monty knows where the prize is and is simply having fun. That is, he will always have his assistant open a door hides a goat. Why is Monty's knowledge crucial? Ah, well. That's a poser. But we'll get to that in future posts.
One last thing. I'm not saying that if you always switch, you will always win. No. You will win two-thirds of the time. If you never switch, you will win one-third of the time.
In a post tomorrow, I'll try to explain why the argument two paragraphs ago is incorrect and will give some additional arguments in favor of switching.
TSU
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