The role of probability in solving the Monty Hall problem

[Jenny A]

*FLOPS DOWN EXHAUSTED *

You're welcome.

[Excited Penguin]

*FLOPS DOWN EXHAUSTED *

You're welcome.

Never say you didn't change anyone's mind.

[The Inquisition]

This thread was very helpful in explaining this, I couldn't quite understand it either.

I found this explanation here:

To me, the clearest explanation is that the only way to get it wrong by switching is to have picked the correct door in the first place. The odds of picking the correct door first are 1 in 3.

https://www.reddit.com/r/AskReddit/comme...ms/c0bkzpj

[robvalue]

Oh yeah. I got suckered by this one first time.

Considering maths is my forte, I felt pretty embarressed.

The important part is the game show host knows where the goat is. He's not randomly revealing information.

Probability remaining door has the car = probability either of the two doors you didn't chose had the car = 2/3

Probability your door has the car = probability you chose right initially = 1/3

The guy opening the door is a kind of misdirection. By switching you're basically choosing doors 2 and 3.

[Fidel_Castronaut]

I see your  problem. You are assuming Monty cheats. Otherwise we know the probability.  Each door has a one third chance of revealing the prize and a two thirds chance of revealing a goat. By revealing a goat door we didn't choose, Monty gives a mathematical reason to switch doors.


It is counter intuitive.  And I remember being really bothered by this problem.  But the math does work.

There is no more parsimonious way than explaining the MH 'problem' than this post. 

You need to look at it in terms of simple probability and the fact that the presenter opens up a door you both (now) know contains a goat.

[Fidel_Castronaut]

The

Oh yeah. I got suckered by this one first time.

Considering maths is my forte, I felt pretty embarressed.

The important part is the game show host knows where the goat is. He's not randomly revealing information.

http://www.coopertoons.com/education/mon...yhall.html

The above website actually goes through the problem with several formats, including a spreadsheet formula. Fun read actually!

[robvalue]

Haha yes, that's exactly what I just scribbled down on my notepad

I agree with Jenny, if you don't believe the maths, go try it out.

If you can code, run a simulation 1000 times with you always switching, and see how many times you win.

If I switch, regardless of what door opens, I win if the goat was behind 2 or 3 initially.

If I don't switch, I win if it was behind 1 initially.

[Chas]

You can find me any proof you like, it still doesn't make sense that probability helps you in that scenario. I will say it's all luck since you can't explain it.

As I already said, the fact that the moderator takes out all the bad variants but for one doesn't tell you anything about whether you made the right decision or not. Just the fact that he takes out all the other doors, doesn't mean that he made the unchosen left variant any more plausible than the one you already chose. And that's because no matter if you were wrong or right at first he was going to take out the same number of doors after you made your choice.


In some cases people refer to the 2/3 logic. That's bad logic. It's not 2/3 anymore because there's no three left. There are only two options left and so it's 50/50, not 25/75.

You are admitting, even bragging about, being ignorant. 

I suggest you study probability because you really do not understand it.

[robvalue]

Indeed.

Consider that your strategy is to switch no matter what. Simple as that. You choose door number 1, and you will switch to the remaining door regardless of which door the guy opens.

How often will you win?

You will win each time the speedboat was initially behind door number 2 or 3. You will lose each time it was behind door 1.

Basically, you're choosing 2 and 3 initially, and the guy gives you one of them for free, leaving the other one.

It is counter-intuitive. Very much so. But he isn't randomly giving you information, that's the point. If he could open your door as well, then the problem would be different. But he can't.

[Jenny A]

One more simple way to look at it. When you choose one door the probability of the door you choose having the car is one third. There is a two thirds chance that the car is behind one of the other two doors. By showing you which of the other two doors not to choose, Monty is letting you take all of the two thirds probability in a single door. So you can either keep your one third or opt for two thirds. It is exactly the same as if you chose one door and then Monty said you could either keep your door or take the other two doors.