The Monty Hall Problem

[denny]

Some friends bent my brain yesterday with a thing called 'the Monty Hall problem' - I thought I'd share the sheer confusion with others...

Description of problem, and ensuing discussion (including lots of explanations which may either help or hurt your head, or possibly both):
http://www.livejournal.com/users/duranorak/790534.html?thread=4120070#t4120070

An off-site explanation, also linked in the above discussion but posted separately here out of sheer helpfulness (I still didn't get it after reading this personally):
http://www.io.com/~kmellis/monty.html

Comments

[wechsler.livejournal.com]

j00 ca|\|n07 c0|\|fU53 m3 \/\/17h j00r g0t35 ;p

[velvetfox.livejournal.com]

Hurray, I get it, maybe I should do that last year of A Level maths one day :)

[feanelwa.livejournal.com]

"The Monty Hall problem.
There are three doors. Behind one of them is a valuable prize; behind the other two is nothing of value. You pick a door. The host of the gameshow opens one of the other doors to reveal that it has nothing of value behind it. Should you change your mind?
(The answer is "yes".)"

(for heaven's sake. We don't all have infinite time to wait for a window to load on a very small network you know ;)

Anyway - have I got this right:
The game show host has been taken to one side and told to do this:
-If [chooser] chooses the correct door, I have to open all other doors but one, the one being picked by random chance.
-If [chooser] chooses the wrong door, I still have to open all other doors but one. But on pain of death I cannot show the valuable thing.

(Therefore the one door must end up as the valuable thing.)

And if I am right, why on earth isn't the bit about what the game show host has been told, put into the question? Seems to me it's one of those situations where the lecturer posing the question doesn't give you all the information in order to look more intelligent because he's scared of the students.

[valkyriekaren.livejournal.com]

Yes - the important part of the problem is that, not only does the gameshow host know he is going to reveal a dud prize, but that the contestant knows it too.

[feanelwa.livejournal.com]

They didn't tell me that - maybe it's my low opinion of TV, but I always start from "the game show host doesn't know whether a given door contains the prize or not until he get to it and find out whether the producer wants him to open it or not".

[dennyd.livejournal.com]

I've never heard JANet referred to as a 'very small network' before. Except by Americans. :)

Yes, you've stated the problem correctly. What perplexed me for ages is why the odds of the other one he didn't open being the prize are higher than the odds of the one you chose being the prize, as they both had equal odds of being the prize to begin with.

Your statement in brackets only applies to the second instance, where the wrong door was originally chosen - in that case the other unopened door is the prize. In the first instance, where the correct door was originally chosen, then obviously the other unopened door doesn't contain the prize. I'm not sure if that's what you meant or not - the way you separated the bracketed statement out from the two cases makes it look like it applies to both?

[feanelwa.livejournal.com]

In the brackets I meant that was what happened if you picked the wrong door the first time. (I never pick the right door the first time!)

I menat you are on a very small network (of one) and I'm sharing the out-cable from college with 400 women looking at Quizilla. Therefore you should cut & paste the problem instead of giving me links to follow! (Although it was at 1:00am and they were all out getting drunk...)

[dennyd.livejournal.com]

Yeah, I wish I'd cut and paste it in retrospect, if only to separate the intuitive solution confusion from the helpful explanations :)

That said, most of the helpful explanations only seemed to confuse the matter more for me. It's quite amusing trying to explain it to someone else once you've figured it out, and watching them look blankly at you and realising you can't explain it comprehensibly either :)

[stuartl.livejournal.com]

Just like the latter link I shall split the problem into two. The first option is when you've selected the right door, the second is when you've selected a wrong door.

You have a 1 in 3 chance of selecting the right door when the problem starts. You have a 2 in 3 chance of getting it wrong but for now we'll assume that luck was on your side. At this point in time you have a 1 in 3 chance of being right, a 2 in 3 chance of being wrong.

The host knows where the prize is, he knows whether you are wrong or right. To confuse you further and increase the drama in the game show he selects one of the other doors. In this situation you're right so he picks one of the other doors at random and opens it. The problem doesn't have 3 doors any more, the problem has 2 doors. One third of the problem has been removed from the equation. This leaves your choice being a 1 in 2 chance of being right. 50/50, no?

Now let's assume that (like the majority of choices) you pick one of the wrong doors. At this moment in time the game show host knows that you've made the wrong choice and selects the one remaining wrong door to open. His choice is not a random selection, it is a preconsidered decision. In opening that door once again that door is no longer part of the probability problem. We've reverted to the situation above. By reducing the options to two doors he has increased your original 1 in 3 odds to 1 in 2.

The important thing to remember is that it doesn't matter whether the game show host has made a considered decision. By revealing one of the three doors the chances have been radically altered from 1 in 3 to 1 in 2.

Now the really interesting thing to me is that if the game show host always opens a door and offers you the chance to switch (by game tradition) then you always have a 1 in 2 chance of getting the prize. This is because you CAN'T select the door that the game show host will open, he will always open one of the two doors you haven't selected. He will always leave your door and one door remaining.

Now the practicality of it all is that if you're serious about it you'll be playing a much more psychological game than a probability game. You will be attempting to deduce from the game show host where the prize is at the point he chooses which door to open.

[dennyd.livejournal.com]

No, the odds aren't 1 in 2 after he's opened a door - because they're a function of the original odds.

If you were right, then his choice of door is random, but if you were wrong, then his choice of door is constrained - which means the original odds have bearing on the secondary odds. Something like this: There's a 1 in 3 chance of you having a 1 in 2 chance of winning by sticking with the original door you chose, and there's a 2 in 3 chance of you having a 1 in 1 chance of losing by staying with the original door you chose. The eventual odds are (I think) 1 in 6 if you stay with your original door, and 5 in 6 if you switch. Although I could be wrong.

Bloody weird problem.

[deliberateblank.livejournal.com]

You have a one in three chance of picking the right door to start with. Once you've picked a door, you have that one in three chance of getting the gold. Nothing you later learn changes the odds of this.

To be able to change the odds of that door from 1/3, you need to have extra information about i) that door you chose, or ii) *all* the doors you didn't choose. But the host carefully does not give you that information (in the usual set of assumptions about the game - the 'correct' solution to the problem hinges on what assumptions you make about the host's behaviour. This isn't being cleverclever, you can get different outcomes with different assumptions but you have to state what they are to justify your answer. With the usual assumptions people still intuitively get the wrong answer.)

What the host has actually told you by his choice of door is "*IF* the gold is behind one of the two doors you did not pick, then it's not behind *that* one." He doesn't say anything about whether the gold is behind the other door you didn't pick or not, therefore he's not told you what you would need to know to decide that the combined probability of the gold being behind either of the two other doors has changed away from 2/3.

The probability behind your door stays 1/3, but the two probabilities behind the other doors have changed from 1/3 and 1/3 to 2/3 and 0/3.

[feanelwa.livejournal.com]

Ah, I see. I'd got as far as the logical bit but not as far as putting the numbers into it.

[stuartl.livejournal.com]

Right. This afternoon I set about writing a C program to prove you wrong.

The program plays the part of the player and the host and does 20,000 iterations of the game using 512 doors in about a tenth of a second. Yes, it's quite nippy, and highly unoptimised because I wanted to be 100% sure the logic was right.

However, halfway through the program I realised why you're all correct. I continued writing the program anyway to prove the point (and yes, it does prove it perfectly) but I saw the bit I'd missed before.

This is how I explained it to myself:

After you randomly choose your door there is a (n-1):1 chance that the prize is in the remaining doors. i.e. It's almost definate.

By the host opening the remainder of the doors he's narrowed down the prize location to the one remaining closed door. As someone else said, the chances of it being in that door are still (n-1):1.

QED, I guess. Apologies for being an arse :)

[dennyd.livejournal.com]

Wish I'd thought of doing that... I argued with people about it for ages before I suddenly clicked :)

[fellcat.livejournal.com]

This is stats/probability math. I therefore consider myself under no obligation to understand it.