The role of probability in solving the Monty Hall problem

[Jenny A]

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.

[Excited Penguin]

I am not assuming the moderator cheates. The math may very well "work"(whatever that means), but in this case it doesn't tell you anything - it's not useful.

I repeat myself, it seems to me like some of you don't even want to know if this method works or not, to you it's merely about understanding how it works. Well, I already understood this particular way to solve it and I'm saying it's wrong.

[Jenny A]

I am not assuming the moderator cheates. The math may very well work, but in this case it doesn't tell you anything - it's not useful.

I repeat myself, it seems to me like some of you don't even want to know if this method works or not, to you it's merely about understanding how it works. Well, I already understood this particular way to solve it and I'm saying it's wrong.

Why?  Why that the math works is obvious.   What makes you think it doesn't?

[Excited Penguin]

I can't prove a negative. You have to prove that it does, not the other way around. And if you proved that it did work, you would have proven a trustworthy pattern and you would have good reason to rely on probability. But until then you don't have any pattern, no proof that it works and no reason to use probability to make your choice.

[Jenny A]

The math is the proof. But if you like, find a friend with time on their hands. Gives them three cards, two jokers and an ace. Have them shuffle and lay out the cards face down. Choose a card. Have your friend reveal a joker from the other two cards. Reveal the third card. Do it a three hundred times. Was the card you revealed an ace more than a third of the time? Chances are very high that it will be.. Try it and see. It's an easy experiment.

[Excited Penguin]

That's just it, Jenny. I shouldn't have to try it - it should be obvious going in and it isn't. In no version of this problem does it say that the one making the choice has played this game numerous times before and that switching was the right thing to do in most of them.

If I did the experiment multiple times I wouldn't do anything but try and find a pattern in order to justify my choice. The problem however is that this math puzzle doesn't involve a variable such as you having played the same game numerous times before and getting a certain outcome most of the times.


And even if I did this experiment you suggested and it turned out that in most cases I won because I switched, that wouldn't tell us anything about any future similar experiment. It just wouldn't. It would only help you with that one particular one you were doing at the moment, presumably(if you can call that kind of luck help).

The math is no proof because it doesn't make any sense as it is.


The math is bad. You can try and prove it however you want but in the end nothing can make one option better than the other no matter how you go about it.


How about you do it instead and tell me what happened. The chances are very high that nothing out of the ordinary will happen and you'll find no reliable pattern.


From what I'm hearing from the other side it seems to me that using probability to make a choice in this case is nothing if not some sort of excuse to feel better about your decision - it's delusional, however. Just because you make some sort of order out of chaos doesn't guarantee you'll win. It's still a chaotic outcome and there's nothing you can do to make it any less so.

[Jenny A]

No you shouldn't need to because the math makes the result obvious. But like all probability problems it can be demonstrated empirically. The more times you play the game the more obvious it will become that switching cards is the right answer. You can show that tossing two pennies results in one head and one tail in 50% of all two coin tosses by making a few hundred such two coin tosses. But the math will tell you that before you start. Similarly, if you play the three card game a few hundred times the experiment will vindicate the math.

[Excited Penguin]

It's not obvious at all. It's wrong in fact. No one has addressed my arguments so far so I'm done responding to straw man ones.

[Jenny A]

Look.  We have a couple members in Portland or Seattle.   If you want to bet, I'll pay you the cost of your plane tickets plus $10,000 if after a thousand tries the switched choice isn't an ace more than one third of the time. If it is, you owe me $5,000.  I could find a use for  $5,000.  But I wouldn’t want to steal your money.  Try the experiment with a friend at home first. You'll never bet me if you do.

And you still haven't voiced any argument beyond your gut distrust of the math. Offering a gamble is the only answer I have to your gut.

[Excited Penguin]

I don't "distrust" math. I think to use math in this case is unhelpful, that's all.


Even if you'll get your ace more than one third of the time you still won't be able to explain why. It will be pure luck.