I was going to go to bed early today, but then I thought of a relatively clear explanation of the Monte Hall problem for those who have grokked the Play-Doh analogy of probability mass, i.e. that you've only got a fixed amount to dole out, and you must dole it all out.
The Monte Hall problem goes like this: Monte Hall presents you with three doors. Behind one of them is a shiny new car, and behind the other two are goats. You are invited to pick a door, with the understanding that if you pick the door with the car behind it, you keep the car. You accept, and pick a door. Monte Hall then opens one of the doors you didn't pick, revealing a goat, and he now asks you if you want to switch to the other unopened door, or stick with your original choice. What do you do?
Now, if you're familiar with this problem, you'll know that, counterintuitively, you should always switch to have the best chance of getting the car. Or if you're really familiar with the problem, you'll know that the correct answer depends very much on what rule Monte Hall is following when he picks a door. On the actual game show this came from, his decision rules are thus:
- He never opens the door you picked.
- He never opens the door with the car.
It's clear enough that when you originally picked your door, you had a 1/3 chance of picking the car, and a 2/3 chance of picking one of the goats. In the Play-Doh analogy, this means you've got 1/3 of your Play-Doh on each door. If Monte Hall follows the two rules above when he opens a door and reveals a goat, then the salient fact to which I'd like to call your attention is that he hasn't given you any new information about the door you picked, but he has given you new information about the other door he didn't open.
Regarding your Play-Doh:
- You have to move it off the door he opened, since there's no longer any chance the car is there
- You can't move it to the door you already picked because he hasn't given you any new information about the door you already picked.
To see that he hasn't given you any new information about the door you picked, consider first the probability that he opens a door with a goat if you picked the car. That probability is 100%. Both doors his decision rules allow him to pick hide goats. Now consider the probability that he opens a door with a goat if you instead originally picked a goat door. It's still 100%, because there's still always one door you didn't pick that contains the other goat. Whichever world you're in, he still opens a door with a goat, so it can't be evidence for or against the car being behind the door you picked. In other words, you still think the door you're sitting on has a 1/3 chance of hiding the car.
But if Monte Hall just forced you to move Play-Doh off of the door he opened, and you can't move it to the door you originally picked, the only place for it to go is to the other door neither of you picked. Which means that door now has twice the Play-Doh on it that you've got on your current door, so you should switch to it.
The reason that feels strange to people is that Monte Hall clearly didn't give you any evidence allowing you to update your probability assignment to the door you originally picked, so it feels like he didn't give you any information at all. But that's false. He implicitly gave you evidence about the remaining unopened door.
If you're having fun, let's take this further. I mentioned that his decision rules are crucial to the question we're considering, so let's weaken them separately to see what effect that has.
First imagine that he's allowed to open the door you picked, but he still never opens the door with the car. The observed data is also still the same; he opens one of the other doors, revealing a goat. Following the same line of thought from earlier, we need our two conditional probabilities:
- The probability he would open a door with a goat if you're sitting on the car door is now 100%, because he has plenty of goat doors to choose from that are not the car.
- The probability he would open a door with a goat if you're sitting on a goat door, however, is 50%. Remember we said he's willing to open your door, as long as it's not the car.
Because these two probabilities are different, the likelihood ratio between them is no longer 1:1 (it's now 1:1/2), so he has given you some information specifically about the door you picked, so some Play-Doh will move to or from that door, so the earlier argument no longer works. (In fact, under these new decision rules and observation, there's no reason to switch. An equal amount of probability mass ends up on your door and the other unopened door.)
Next, imagine that he's allowed to open the door with the car, but he'll still never open the door you picked. He still actually ends up opening one of the doors you didn't pick, and it's still a goat. Again following the same line of thought from earlier:
- The probability he would open a door with a goat if you're sitting on the car door is 100%. The two doors he's allowed to pick both hide goats.
- The probability he would open a door with a goat if you're sitting on a goat door is 50% like in the last analysis, but this time for a different reason: In this scenario he's not willing to open your door, but the two doors he's allowed to pick contain a goat and a car, so it's 50% he'll open the goat door.
Again, the probabilities are different, so the likelihood ratio between them is not 1:1, so the original argument doesn't work. He has still revealed equal information about the door you picked, and the final remaining door, and the result is equivocal.