Was Monty Hall a Bad Statistician?

You may have heard of the Monty Hall problem.^[1] If you haven’t, it’s a bit of a brain teaser based on an American TV show, named after its host, Monty Hall. The problem goes:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

I'm not going to get into why it is (Wikipedia will explain it better than I), but the commonly accepted answer to the problem, and what’s touted as the statistically optimal choice, is always to switch doors once a goat is revealed.

Let's think about this brain teaser a bit more deeply, though. Why would Monty Hall give a statistically optimal option to punters that beats the stated 1/3 odds of the game? Was he just a bad statistician?

It turns out in reality that the Monty Hall problem is a bit of a silly example. If Monty is constrained that he must open a door with a goat, and that he must allow you to switch doors, a rational contestant will win the car 2/3 of the time.

In reality, when the game show was ongoing, Monty had discretion both over whether to open a dud door and whether to offer a switch.^[2] In fact, if you played the "always switch if allowed" strategy, Monty could outplay you by only offering the switch when you had already chosen the door with the car. The house always wins.