- 23 Jul '07 22:54 / 1 editThe monty hall problem is one of the most talked about problems in history. It is based upon the show "Let's make a deal," in which their are three doors, and behind one is a prize, such as a car, and behind the other two doors are goats. In this show, a contest would choose any of the three doors, (lets say door number 1). Rather than revealing the candidates selection, Monty would instead open up another door (lets say door 3) and show a goat. Monty would then ask whether the contestant would like to stay with their selection (door 1) or switch to the other door (door 2).

Question: After Monty Hall shows that one door has a goat, decide whether it is a better Idea for the candidate to stick with their initial selection, switch to the other door, or if it is just doesn't make any difference at all. If you can figure it out, then explain how you got your answer. It took me a while, but I eventually figured it out. - 23 Jul '07 23:32 / 1 editIt is better to switch to the other door, as you will have a higher chance of winning the prize.

If you stick with your initial door, the probability of winning is 1/3. However, if you switch to the other door the probability of winning is doubled to 2/3.

This result is counterintuitive, as when the presenter removes one door most people assume that there is an equal chance of finding the prize behind each of the remaining doors. This is not the case, as the presenter will only ever open a door which does not contain the prize.

Consider the initial choice of door. There are 3 doors, all equally likely to contain the prize. One is choosen at random, and so has a 1/3 chance of winning. Now the presenter opens a door which does not contain the prize. Now, there is still a 1/3 chance that the door you choose originally contained the prize, so there must be a 2/3 chance that the door you didnt choose contains the prize.

This can be explained in a more intuitive way. There are three outcomes (the doors have been labeled "winning", "goat A" and "goat B" ):

1) The player picks the winning door. The presenter now has to open either goat A or goat B.

2) The player picks goat B door. The presenter must now open the goat A door.

3)the player picks goat A door. The presenter must now open the goat B door.

In 2) and 3) switching doors will result in a win, in 1) switching doors results in not winning.

So out of the 3 possibly initial choices, 2 of them result in a win if the contestant switches, and only one results in a goat if the contestant switches. There is a 2/3 chance of winning if the contestant swaps, compared with a 1/3 chance of choosing the winning door initialy. - 23 Jul '07 23:38Yes, I can imagine. It is an easy assumption to make, as when you choose a door all 3 have the same chance of winning, so if one is then removed many people will think that both doors still have an equal chance of winning.

The trick is realising that the presenter will never open a winning door. A tree diagram reveals the correct answer at a glance though. - 24 Jul '07 16:31

ON WHAT BASIS?*Originally posted by doodinthemood***The way you've phrased it, it is better to switch.**

If instead you make the question experience based, and you are in the situation having never watched the show before, it is better to stick.

That is an asinine comment. Do we assume that everyone in this position is unable to work out the probabilities!?!

Crap comment. - 24 Jul '07 17:09No, but if you find yourself in that situation, you are not aware of whether the host opens the doors every time. You have only experienced one possibility, whereby he's opened a door this time. Had you picked an incorrect door, it seems reasonable to accept that if he doesn't open another door every time, he would have let you open your own door this time. Instead, by opening a door, he indicates a will for you to change. Thus you should stick.

It may be hard to think like this, which is an often exploited defficiency by magicians. There are three piles of cards, he tells you to take one. You take one, he puts it to one side "ok, we'll take that away then. Now, I get one of these packs and you get one. Which one do you want me to have?" you point at one. He says "ok, so you take away the other, and I'm going to keep this pack". He then shows that you've limited it down so the magician keeps the only pack which isn't blank! OK, that's a rubbish trick, but you get the idea. You haven't seen the situation whereby you pick a different pack at the start, or in the second stage, so you have to judge on hypothetical decisions. - 24 Jul '07 18:51

Your reasoning is garbage... the situations are not the same at all. If Monty Hall has opened the door, then the situation is determined already, and logic can be applied. It is easily assumed that he won't open the prize door, because then when he asks you to change you would pick the prize. If he opens a door and doesn't ask you to change, then you are never in the position to make a decision anyway, so the point is moot.*Originally posted by doodinthemood***No, but if you find yourself in that situation, you are not aware of whether the host opens the doors every time. You have only experienced one possibility, whereby he's opened a door this time. Had you picked an incorrect door, it seems reasonable to accept that if he doesn't open another door every time, he would have let you open your own door this ...[text shortened]... t the start, or in the second stage, so you have to judge on hypothetical decisions.**

And the card trick thing is used extensively, and can be used multiple times to the same audience by just saying, "I'm doing another trick." And they would believe you. Stop with the magic example it sucks and is not relevant. - 24 Jul '07 19:08The example is very relevant. It emphasises the fact that you shouldn't assume this situation would be reached had different choices been made in the past. Had you picked an empty door, would the host have any need to give you the possibility to switch? no. He would have let you open your own door straight off and lose. Only if you pick the correct door would giving you the option to switch be a necessity to provide the possibility of you losing. You should stick.
- 24 Jul '07 21:06

Didn't you try this little exercise in changing the parameters of the problem the last time someone posted a Monty Hall type problem? If you want to post your own version of the game, with a set of rules that make sense, then do that, but stop hijacking other people's threads for this nonsense.*Originally posted by doodinthemood***The example is very relevant. It emphasises the fact that you shouldn't assume this situation would be reached had different choices been made in the past. Had you picked an empty door, would the host have any need to give you the possibility to switch? no. He would have let you open your own door straight off and lose. Only if you pick the correct doo ...[text shortened]... the option to switch be a necessity to provide the possibility of you losing. You should stick.** - 24 Jul '07 21:24The Monty Hall problem is a very well defined problem. And the solution is also very well defined.

But, if I recall right, Monty himself altered the rules, sometimes ( sometimes not), to something that was not the true original problem. People got angry and called Monty a cheater. But who would argue about it? It was Monty's show after all, he changed the rules as he wanted.

But when we're talking about the true Monty Hall's problem, it is really very well defined and is used in probability class the world around. - 25 Jul '07 08:08

I think I may have done. It may have been on a different forum but I cannot really remember. And as there's pretty much nobody who isn't aware of the 66% switch answer nowadays (which is indeed correct for the wording given on this particular occasion) I feel the near100% stick possibility is worth explaining to cast some new light on the problem.*Originally posted by BLReid***Didn't you try this little exercise in changing the parameters of the problem the last time someone posted a Monty Hall type problem? If you want to post your own version of the game, with a set of rules that make sense, then do that, but stop hijacking other people's threads for this nonsense.** - 25 Jul '07 08:44

Yep. That was him.*Originally posted by BLReid***Didn't you try this little exercise in changing the parameters of the problem the last time someone posted a Monty Hall type problem? If you want to post your own version of the game, with a set of rules that make sense, then do that, but stop hijacking other people's threads for this nonsense.**