The role of probability in solving the Monty Hall problem

[Excited Penguin]

I just read a post on WaitButWhy's blog and it's about a variation of the original Monty Hall problem that goes like this:


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?

Source - Monty Hall Problem

My contention is that you can only rely on probability in a controlled environment where you have knowledge of a certain definite pattern of choices that will always hold true(id est, e.g., you would have to know that out of 300 hundred such cases option one will be chosen one hundred times, option two one hundred times, and option three one hundred times as well and that this pattern would hold true no matter how many cases you would observe. . Then it would be logical to rely on probability, but otherwise, you have to keep in mind that for you the choice is totally random no matter what anyone does - unless someone actually tells you which one is the right one and you have 100% reason to trust them.

What would you think and can anyone convince me why it's still better to go with probability even if you don't have such(^^) knowledge?


For those who are interested in better understanding why one would think it's better to switch.
WaitButWhy: http://waitbutwhy.com/2016/03/the-jellyb...oblem.html

For anyone interested, here's the same argument I made on reddit about it, but talking about WBW's version of it: 
https://www.reddit.com/r/WaitButWhy/comments/49es04/the_jelly_bean_problem/


----About the poll, some would argue that the I don't know option is just a variation of No when you get right down to it(at least in this particular case), but I put it in anyway for those of you that don't see it like that.

[TheRealJoeFish]

Here's the easiest way I've found to convince people why switching is better:

Suppose instead of three doors, you have 100. There's a car behind one and goats behind 99 of them. You get to choose a door, and then 98 other incorrect doors will be eliminated.

So, you choose door 34. The host opens eliminates 1, 2, 3 ... 33, 35, 36... 85, and 87 through 100.

So either the prize is behind door 34, which you picked right off the bat, or 86, which is the only one the guy didn't open.

Intuitively, it's clear that they don't have the same probabilities of being correct, right? Essentially, you've always had a 1% chance of it being in 34, and the other 99% collapses into 86.

[Excited Penguin]

Yes, Why But Why explained it like that as well, I've already heard it. And that would be a good argument if I believed it best to rely on probability in this case, which I don't and I explained why.

I wouldn't switch not because of one option making more sense than the other, but purely because of my intuition - since I don't think reason will help you in that situation at all.

[TheRealJoeFish]

Umm. Ok wel then I don't know what to tell you. You're essentially saying "the math tells me my expected earnings will be greater if I switch, but I'll go with my gut."

I don't even know how to respond to that.

[Excited Penguin]

It's that math doesn't tell me anything in that case(doesn't help/doesn't work/doesn't make sense), not that it tells me something and I choose to ignore it. And it's particularly dishonest of you to employ that sensibly false analogy.


Let me put it like this. What if you live in an universe where 100% of the time the correct option will always be the one you chose to go with at first. How will probability help you then? How does probability tell you that you don't live in that universe?


I think probability is only helpful when dealing with patterns. There is no pattern to be observed here, you're on wholly new territory with this problem and you don't have much of anything to go with. For example, you have no reason to assign any probability to any of the three options. To think that you do, I think, is simply so because of associating this particular scenario with others in which you would have more knowledge about what you're dealing with.

[TheRealJoeFish]

Ok now you're being weird. Of course the math tells you something. That's... what math does.

The definite pattern that you suggest doesn't exist is: "one door has a car and two doors have a goat."

Like... I just don't even

[Excited Penguin]

Your problem is, it seems to me, that you're taking the math for granted despite the fact that you can't properly defend its use in this case. That's both dogmatic and unreasonable. I'm challenging the notion of using probability to deal with such a scenario and you didn't come up with any good reason why I should use it.

It seems to me that unless you study a multitude of such cases and switching because of probability turned out to be the right thing to do in most of them you couldn't possibly know that it would work. And even if that were the case and you knew it you still wouldn't be able to explain why it did. But the thing is you don't even have that kind of knowledge going in, in fact you have none. Therefore going with probability is illogical.

[Jenny A]

What is it about the Monty Hall problem that makes you think math doesn’t apply? It looks like a straight up probability problem to me. Therefore it's a mathematical problem.

If you are assuming Monty cheats then there is no right answer because Monty will win if he wants to.

[Excited Penguin]

What is it about the Monty Hall problem that makes you think math doesn’t apply?  It looks like a straight up probability problem to me.  Therefore it's a mathematical problem.  

If you are assuming Monty cheats than there is no right answer because Monty will win if he wants to.

It's that you have no reason to go with probability when having no knowledge of the variables involved. As I already said, probability is only helpful when you spot a pattern and there's no pattern here, therefore probability doesn't help you at all.

I'm not assuming anything. That's just it, I can't assume anything so I don't. But those that would make a choice because of probability would be assuming things.

[Excited Penguin]

Lets say you have a billion options. You make a choice and the moderator takes away 999.999.998 of them, thus leaving you with the one you chose and one other one. I think it's an illusion to think that this alone makes the other one any more likely to be true than the one you already chose, even if the ones he took out of the equation are all false. And this is because no matter what(that is, even if you made the right choice to begin with) the moderator is going to take away 999.999.998 of them away. So just the fact that you "made it this far" and that all of the ones he eliminated are the "bad ones" doesn't tell you anything. And that's because the game wouldn't have ended even if you made the right choice at first - and you know this. No matter what you chose, right or wrong, the moderator is going to take away the same number of options leaving you with a 50/50 chance of being right in the end. To assume that now it's more likely you're wrong is to assume that the moderator would end the game if you made the right choice at first, and that's a wrong assumption to make.