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Notes -
Doesn't the feeling go away when you realise that what you're choosing is the odds of selecting the car the first time around, not the second, since the game show master always removes a goat? The removing of a goat isn't independent of your choice of door.
Or if you expand the number of doors. If there are a million doors, you choose one and the game show master removes 999998 doors and asks if you want to choose again, do you still feel like it doesn't matter what you choose?
I was never convinced by the "expand the number of doors" scenario because it's not obvious why 999998 doors would be removed in the new scenario. It feels like another euler trick.
Yeah me neither because of how hand wavy it is. It's trying to make a point about priors but that totally misses the point! The point of the Monty hall problem is that the door Monty Hall opens is DEPENDENT on you having made the choice before. That's what makes it 2/3. If he already opened a door and you walked up to the stage and chose the door after the fact, it would be 1/2. The lesson in Monty hall is a lesson of dependence not priors.
Another way I like to view it is that Monty Hall opening the door doesn't tell us anything at all about the door we initially chose. It's the illusion of information, my chances were 1/3 when I walked up to the stage, it's still 1/3. Because the dependence flows one way. You are not actually given any information to update on, so I think the bayesian explanation is kinda shoddy on that front.
Anyone who is actually good at probability theory, feel free mansplain it to me If my intuition is wrong.
Yeah, you’re right. When you pick door 1, Monty removes it from his little pre-game winnowing of the doors; he can only open door 2 or 3 now, which collectively have a 2/3 chance of containing the prize. If he opens 3 and it contains the car, you’ve automatically lost. The chance of this is eliminated when he reveals a goat, but the collective grouping of 2 and 3 retains 2/3 odds. Now door 1 is added back to the mix, it’s smart to switch.
What is sometimes confusing is realizing that there are actually two ‘games’, each with only two doors in play, with the second dependent on the first. The odds of a door having the car in a standard two-door scenario are 50/50, but the odds of your door having it are only 1/3, because there’s a 1/3 chance that you’re only still playing because Monty excluded your (goat) door from round one. Your door being ‘safe’ from Monty’s initial opening means your odds don’t improve, while door 2’s do, provided 3 doesn’t have the car.
When you frame it in terms of dependence (on you choosing the door before or after Monty showing a door with a goat) the Monty Hall problem isn't much of a problem at all.
This is a common thing in many many stats puzzles and the overall trend of why normies are so bad at probability/stats. The most important/pivotal part is formulating the correct problem statement. And in terms of inference, that means knowing exactly what the inference tells you and what it doesn't.
The above applies to most branches of math, but probability/stats is fundamentally at a level of abstraction above most other fields of math from the ground up.
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