this post was submitted on 26 Mar 2024
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The Monty Hall problem is not a paradox, and I'm hesitant to call it a conundrum. It has a very simple solution. The "point" of it is that people inherently don't like that solution because it challenges their instinct to stick with their first choice.
Correct, extend it to 10 or 100 choices instead of 3 and it's easy to see.
Me: Pick a number between 1 and 100.
Them: 27
Me: Okay, the number is either 27 or 44, do you want to change your choice?
Them, somehow: No, changing my choice now still has the same probability of being right as when I made my first choice.
It's obvious that they should want to change every time.
That’s a great way to look at it. I’d just call it ‘counterintuitive’ in the Monty Hall formulation.
I: 27
You: The number is either 27 or 44. Do you want to change your choice?
I: why would I?
Because when you first picked 27, it was 1 out of 100 choices. Then I tell you that you either got it right, or it's this other number. None of the others are correct, only 27 or 44.
So you think your 1/100 choice was better than the one I'm giving you now? On average, you'll be right 1% of the time if you don't switch. If you do switch, you'll be correct 99% of the time.
Another way to think of it is: you choose 27 or you choose ALL of the other 99 numbers knowing that I'll tell you that 98 of them are wrong and you'll be left with the correct one out of that batch. One of those clearly has better odds, no?
In this example, there were 100 choices in the beginning, and later you reduced to 2 choices. Clearly an advantage. Does the same apply to the 3 door problem?
Let's take this question in another angle. Instead of 3, there are only 2 doors. I am to choose one out of 2, which has a prize. After I choose one, you show me a third door which is empty. Now, should I change my option?
Yes, it's the same concept. The same math/logic behind it doesn't change. You're choosing 1/3 or you are choosing 2/3 and I'll tell you which of the two is incorrect. It's just easier to visualize with 100 doors instead.
I'm not sure I'm following the other angle..there are 3 correct possibilities at the start but I can only choose 2? Or there are 2 possibilities and then you introduce a 3rd door that is never correct?
Yes that one. Similar to the one you did with 100 doors, just in opposite direction.
Do you know the third door is never correct? Because then the probability doesn't change.
Scenario 1: You chose 1/2 at first with a 50% chance of being correct, I introduce a 3rd door (but it isn't a legit possibility), so the actual choice for you is still 50/50 (between doors 1 and 2)
Scenario 2: If you think it's possible that 3 could be correct (but it actually never is) then, no, you wouldn't want to switch. By staying with your first choice has a 50% chance of winning, by switching it only has a 33% chance. But there's no way to know this ahead of time (because as soon as you know you shouldn't switch bc 3 is the wrong door, then you're back in scenario 1)
Scenario 3: For completeness, let's say the 3rd door can be correct sometimes. Then it doesn't matter if you switch or not. It's a 33% chance of winning either way. If there is a chance it can be correct, then your first choice doesn't matter at all and the second choice is the 'real' choice bc that's the only time you're able to choose from all real possibilities.
The only reason that the Monty Hall problem changes probability in the second choice is because you are provided more information before the switch (that the opened door is absolutely not the one with the prize)