I know this topic has been discussed sometime ago on this site – but I offer my take on it, as a proof of the correct solution without resorting to studying conditional probabilities or other complex theories.

To those who have never encountered the problem, it is a worthy problem that has stumped many – including myself. Even when informed of the answer, many still refuse to accept it.

In essence the problem is this – you are the winner on a TV game show, but to win the star prize you have to select the winning door, behind which is the star prize - from a choice of three doors.
Once you have announced your chosen door, the TV show’s host (who knows behind which door the star prize is located) opens one of the other doors, showing you that it is a loosing door – and then gives you the option of changing your chosen door.

The question is does changing your chosen door improve your odds of winning the star prize (or does it make no difference)? In other words, should you swap from your chosen door (which you do not know what is behind it), to the door that the TV host has not revealed what is behind it?

Before you read the solution (below) – give it some thought, and decide whether you should change your chosen door to improve your odds of winning.

Wikipedia has a large article on the subject – after reading it, you might be none the wiser (except to have been informed of the answer).

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The answer:-

OK, let’s imagine 30 people are winners, and each has the choice of choosing any one of the three doors A, B or C.
On average, 10 people will choose door A, 10 door B and 10 door C.

If none of the 30 change their chosen door, and if we assume that the star prize is behind door A – then on average 10 of the 30 will have won the star prize.

Now if we consider that all 30 exercise their right to change their chosen door; then all those who selected door A will be shown door B or C, and so will swap away from their winning selection.
However all those who chose door B will be shown door C, and so will select the winning door A; and those who chose door C will be shown door B, and so will also select the winning door A.

So of the 30 contestants who did not change their selection, will on average win 10 star prizes between them; whereas of the 30 contestants all of whom changed their selection will on average win 20 star prizes between them.

And so the answer to the problem is that you should change your selection to improve your odds of winning.

 

Your explanation is completely valid and may convince some people, but others may get confused because by equating what happens to a group of people to what happens to a single person who has to choose a single door.

The explanation I have found that seems to work the best is to point out that, if you always switch, then the only way you can lose is if you picked the right door initially and the change of you doing that is only 1/3. Hence the chance of winning is 2/3. Equivalently, if you choose the wrong door initially and then switch, you will win. Since your odds of picking the wrong door initially is 2/3....

For people that still insist that regardless of which door you pick, after Monty opens a non-winning door you are left with a 50/50 chance, simply change the scale of the game. Make it 1000 doors with a single prize. After you pick a door, Monty will open 998 doors and then give you the chance to switch. Which is now more likely -- that you picked the 1 door in 1000 that had the prize, or that Monty opened all but the 1 door in 999 that has it?

 

Only if the prize is behind A.

 

Only if the prize is behind A.

If the star prize was behind door B, then of the group who did not change their selection, on average 10 persons would win the prize (those that selected door B)

Of the group that all changed their selection, those that chose door A would be shown door C and so win by selecting door B, those selecting door B would be shown doors A or C and so loose; those selecting door C would be shown door A and so win by selecting door B; so on average 20 persons would win the prize (of those changing their selection).

If you are unable to work out what happens if the star prize was behind door C – this problem is beyond your comprehension.

 

Your premise is only true if all the people act normal. It's conditional. All math is.

That's just my take. I have no mathematic proof for it. Mathematics is not evidence to me.

 

Mathematics is not evidence to me.

Then how the heck did you ever survive in a technical field?

 

So it's really quite simple.
You chances of having picked a winning door are initially 1/3.
But once it's reduced to two doors, then chances of the prize being behind the other door has to change to 2/3 since the chance you've picked it is still 1/3 and the total probability must be 1.
Thus you obviously should always switch doors for the best chance of winning (2/3).

It's not conditional and has nothing to do with whether people act normal or not.

 

It should be abundantly clear from my analysis of the problem, that of those choosing not to change their chosen door have a 1 in 3 chance of winning the prize.

But for those who change their chosen door, if they initially choose either two of the loosing doors, they will be forced to select the winning door – only to lose if they initially chose the winning door.

As stated in the original post ‘Even when informed of the answer, many still refuse to accept it.’

 

Here's and interesting story about the Monty Hall Problem when Marilyn vos Savant posted the correct answer in her Ask Marilyn column in 1990.
Thousands wrote in, including numerous academic PhD's, insisting she was mistaken and totally inept at probability theory, some even insulting her about using "female logic".
Eventually they all had to remove the foot from their collective mouths and eat crow.

 

I do agree with the solution. I have trouble with the name of the puzzle--Monty Hall. The game director did NOT always open one door; he did so only occasionally. What made him choose? Knowledge that the constant had won and a desire to harm him? Knowledge that the contestant had lost and a desire to help him? Maybe even Monty did not know. But if not every contestant gets the same chance, then the analysis using odds does not hold. As a puzzle it's great, as advice in a similar situation, not so much.

 

I do agree with the solution. I have trouble with the name of the puzzle--Monty Hall. The game director did NOT always open one door; he did so only occasionally. What made him choose? Knowledge that the constant had won and a desire to harm him? Knowledge that the contestant had lost and a desire to help him? Maybe even Monty did not know. But if not every contestant gets the same chance, then the analysis using odds does not hold. As a puzzle it's great, as advice in a similar situation, not so much.

Exactly. There's an ambiguity in the usual formulation of the problem (as in the original post) in which it is not stated whether the host always opens a non-winning door. If he does, then switching is indeed the correct choice; but if the host has the option of not opening a door, the probabilities change. If the host is more likely to open a non-winning door when the contestant has chosen the correct door, then obviously switching is sub-optimal.

 

If the host is more likely to open a non-winning door when the contestant has chosen the correct door, then obviously switching is sub-optimal.

Yes.
In that case you would need to know the chances of the host opening the door when you have picked the correct door to know the odds of whether you should change your pick.

 

Here's and interesting story about the Monty Hall Problem when Marilyn vos Savant posted the correct answer in her Ask Marilyn column in 1990.
Thousands wrote in, including numerous academic PhD's, insisting she was mistaken and totally inept at probability theory, some even insulting her about using "female logic".
Eventually they all had to remove the foot from their collective mouths and eat crow.

What I found interesting about the whole Marilyn vos Savant episode was that many people were talking at cross purposes because they weren't actually talking about the same problem. The problem that vos Savant was talking about -- and the problem that nearly everyone means when they talk about the Monty Hall Problem -- has a strict rule that the host MUST open a non-winning door and then MUST give the contestant the opportunity to switch. Many (not all) of the people that took issue with her were taking issue with those premises. There were some that explicitly acknowledged that according to the rules inferred by her column, her answer was correct, but that the rules WERE inferred and there was nothing in the actual conduct of the game show to support that those were the rules. Much later, Monty Hall himself stated that there were no such rules that he followed and that he was free to choose whether to open a door at all or whether to give the contestant an opportunity to switch.

 

I would guess that initially, the process was a means of slowing the game down when necessary. To speed it up all he would have to do is skip some of the banter. I think this puzzle has worn out it's welcome.

 

I would guess that initially, the process was a means of slowing the game down when necessary. To speed it up all he would have to do is skip some of the banter. I think this puzzle has worn out it's welcome.

Oh, it will probably never wear out it's welcome. For most people the result (of the simplified puzzle as it's usually presented) is so nonintuitive that it can force people to better understand many of the common misconceptions of probability and also serves to underscore that humans, in general, do NOT have a good innate grasp of probability.

What I found interesting when I first tackled this problem was how easy it was to rigorously prove that switching doors doesn't make a difference and it can take quite some time to realize the flaw in the proof -- which then makes it quite entertaining to present one of those "proofs" to someone that accepts the correct answer and see if you can convince them that they were wrong and that switching really has no effect. Quite a few people fall into that category because many people (particularly people like students in the sciences or engineering that have grown accustomed to the math not lying to them) will accept a superficially sound mathematical proof without looking closely at the validity of each constituent step.

 

This is, in my opinion, one of those things that depends a lot on the wording and definition of the terms, as well as human psychology. It is not only a mathematical dilemma, but also has to do with how terrible people's intuition is. In addition, it may have a different answer based on your interpretation of the words.

 

Here is my easy way to see the solution by bending the problems bit.

See, opening a door, while exposing additional information, does not change the odds of you selecting the prize.

Instead of opening doors allow the contestant to switch from the one door they chose to all the other doors, and if the prize is behind any door they win it.

Obviously the odds of not choosing the prize first time are greater than choosing correctly. Thus by switching to all the other doors you must increase your odds of winning, excepting the case when there is only two doors.

 

What you would do if you were in a real gameshow is sweat, panic and apply all the cognitive bias you can muster.
Psychopaths exempted of course!

 

Its only a game!!

 

Its only a game!!

But it's a game that involves money.