The "Three Doors" mathematical problem

[Rolfe]

My "hundred doors" example was to show that it did matter whether or not Monty was deliberately avoiding the car.

If you chose at random, you'd have a 1 in 100 chance of having got the car. Assuming you didn't, and Monty was opening doors at random, the overwhelming probability (98 out of 100 I think) is that he would reveal the car at some point during all the subsequent door-opening.

If the car is still hidden when only two doors are still shut, then what do you think? Either you hit lucky with your original guess (1 in 100), or Monty by chance has left the car till the last door (another 1 in 100), or he is deliberately avoiding the car. The last is by far the most probable scenario, and it's intuitively obvious that you'd switch.

The other example I had was also to show if he is avoiding the car. Just run the three-doors trial a number of times. 33% of the time Monty should reveal the car, if he's just opening either of the doors at random. In that case, switching won't help you win more cars, but it won't spoil your record either - it's a 50/50 shot. If Monty never reveals the car, he's deliberately avoiding it, and in that case you benefit from switching by a 1/3 to 2/3 ratio. So the answer is, switch. If he's not avoiding the car, then it makes no difference, if he is, then you improve your odds. You switch to take advantage of the latter case if it applies, as there's no down-side if it doesn't.

In the online version quoted by the earlier poster, it's obviously set up so that Monty never reveals the car.

Rolfe.

 

[garys_2k]

IIRC there were several variants of the game. Sometimes one of the prizes was "Monty's Cookie Jar," which contained an unknown amount of cash. It could be a goat (very little money), or the grand prize, and he would reveal the fact that that jar was behind one of the doors from time to time. I do remember him once showing that the revealed jar was the grand prize.

Another variant had three prize levels. One would be the goat, the other some fairly nice furniture (mid-level prize), the other the grand prize. He would sometimes reveal the goat or one of the more valuable prizes, but I can't remember if he'd ever reveal the grand prize or just the mid-level one.

 

[Matabiri]

If the host reveals 100 doors with goats, then by changing your choice, instead of getting the 33 cars you chose right the first time, you get the 66 (67) cars you DID NOT choose right the first time.

Edited to add: I am not sure what you actually meant by your example, that the odds go up or that they don't?

That's exactly what I meant. It was supposed to demonstrate that swapping is the right strategy. I.e. that the odds don't change.

 

[CurtC]

Rolfe wrote:
If you don't have that information, then overall it's still best to switch. If he isn't deliberately avoiding revealing the car, then you still sit on the 50/50 odds, so it's not going to make your chances any worse.

Given that there is some chance that he is deliberately avoiding the car, and in that case your odds are 2/3 in favour of switching, then always switch. You won't decrease your chances, and depending on the game he's playing (which you don't know for sure), you may increase your chances.

I'm not sure that I understand your argument, but it's easy to imagine a scenario where switching is the wrong strategy. Let's say that Monty only offers the switch if you guessed the correct door already. In this case, you lose 100% of the time by switching.

 

[Number Six]

If you don't have that information, then overall it's still best to switch. If he isn't deliberately avoiding revealing the car, then you still sit on the 50/50 odds, so it's not going to make your chances any worse.

Given that there is some chance that he is deliberately avoiding the car, and in that case your odds are 2/3 in favour of switching, then always switch. You won't decrease your chances, and depending on the game he's playing (which you don't know for sure), you may increase your chances.

Rolfe.

If his choice of whether to reveal a goat to you is not independent of your initial choice then his choice of whether to reveal a goat to you depends on whether your initial choice is correct. And if his choice of whether to reveal a goat depends on whether your inital choice is correct, you would have to know _how_ it depends on your initial choice to make an informed decision.

Suppose you don't know he's going to reveal a door and then you pick Door A and then he reveals that behind Door B is a goat. Should you switch? We can't say. Maybe he was only going to reveal a goat if your initial choice was correct, in which case you should not switch. Maybe he was only going to reveal a goat if your inital choice was incorrect, in which case you should switch. But the point is, if you don't know then you can't say which course of action is best.

 

[Rolfe]

Maybe he was only going to reveal a goat if your initial choice was correct, in which case you should not switch.

OK, I see, it's another variant once you've recognised that Monty's motives are important.

The way I was originally told it, was that he always opens a door and invites you to switch if you choose. But if you don't know that one for sure then yes, it's moot.

Well, I think we've successfully trashed the theory that it doesn't make any difference whether Monty knows what's behind what door, or has any motivation to his actions!

Rolfe.

 

[Interesting Ian]

BillHoyt, you're right that it comes down to whether Monty Hall always shows you a goat-door. But I watched the show lots of times, and the fact is, he didn't. Sometimes, he would just reveal the door that the contestant had picked, right or wrong. Offering the switch was *not* something he did every time.

So now, you have to figure in his motivation in giving you this choice. Is it done to keep you from winning? In that case, never switch. Is it done to make the show more exciting? Then it probably is 50-50. Is it done randomly? Switch. Is it done to give away more merchandise? Switch.

If anyone denies that the motivation of the host is what makes the difference, consider this scenario. You're walking down a city street and come upon a street hustler offering you a game of three-card monte. You put your money down, and the hustler moves the cards around thoroughly, at a speed you can't keep up with. You pick your card. The hustler, instead of revealing the one you picked, turns over a non-winning card, and offers you the choice to switch your guess to the remaining one. Should you switch? If you do, you're a fool.

Ummm . . no.

First of all if he doesn't offer any switch then there is no further discussion.

So let's consider the scenario where he does offer a switch. Either the host knows or doesn't know which door the prize is behind.

Let's consider he doesn't know and consequently reveals the door with the prize behind. Obviously you switch to that door (if you're allowed!).

Let's consider he doesn't know and consequently reveals the door with the goat behind. It is very clear that it doesn't matter whether the contestant switches or not. You have half a chance of sticking or switching to the other unopened door.

Now let's consider the scenario where the host does know. He reveals the door with the prize behind. Obviously you switch to that door (if you're allowed).

Let's consider he does know and consequently reveals the door with the goat behind. Now this is the complex one. If the host always uses his knowledge of what lies behind the doors to choose the door with the goat behind it, then obviously it will be a very wise idea for the contestant to switch, because the probability goes up from 1/3rd to 2/3rds of a chance of winning the prize. But if the host, despite his knowledge of what lies behind the doors, just picks a door arbitrarily (no preference), then again switching will be a half of a chance.


But in whatever scenario, switching does not make you a fool because switching can never make it less likely you would win than if you had stuck with the original door.

In fact we would not know if the hustler knows which door the prize is behind. So all we can say is that by switching the probability of getting the prize will be from 50% - 66.7%. Determining the precise percentage will be a question of psychology.

 

[Interesting Ian]

That is it. It is best to switch _only if_ Monty's choice to reveal a door to you is independent of whether or not you picked the right door. If you know beforehand that Monty is going to ask you if you want to switch, then you switch. But if you don't know that he's going to ask and then he asks, you can't say whether it's best to switch or not because you don't have enough information to come to a conclusion.

No, it's always best to switch because although it might only be 50/50 chance whether you switch or not, the point is you don't know that. The chance of winning the prize is anything from 50% to 67% if you switch. Since you cannot determine the precise percentage, it is wise to switch.

 

[Rolfe]

But in whatever scenario, switching does not make you a fool because switching can never make it less likely you would win than if you had stuck with the original

Unless, as Number Six just pointed out, Monty only offers you the opportunity to switch if he knows you've already picked the winning door.

Once you allow Monty's motives to come into consideration, as you must, you do face that one too unless it is specifically excluded from the original scenario.

However, if the switch is always offered irrespective of whether or not the winning door was initially selected, then yes, that statement above is correct.

Rolfe.

 

[Interesting Ian]

Unless, as Number Six just pointed out, Monty only offers you the opportunity to switch if he knows you've already picked the winning door.

Once you allow Monty's motives to come into consideration, as you must, you do face that one too unless it is specifically excluded from the original scenario.

However, if the switch is always offered irrespective of whether or not the winning door was initially selected, then yes, that statement above is correct.

Rolfe.

Damn damn damn! Just typed out my detailed reply and my computer restarted all by itself just as I finished! Months since it's done that. Maybe bitTorrent got something to do with it. Can't be bothered to type out all again. Anyway . .


If he offers the switch then this makes it more likely as a general policy that either he always offers the switch or more often offers the switch than otherwise. This then still makes it a wiser choice to switch (notwithstanding that reasoning from a single case is dodgy!).

But you're right. If people are familiar with this problem then they will have a propensity to switch. The host can use this knowledge and only offer the switch when you originally pick the correct door. Damn!

Unless you know that he's going to think like this then deliberately not switch when offered LOL All psychological games

 

[CurtC]

Rolfe wrote:
The way I was originally told it, was that he always opens a door and invites you to switch if you choose.

I've been hearing this puzzle for about 15 years, and I don't think I've ever heard it explicitly stated that the host is required to open a door. The puzzle is solvable only if that stipulation is added, and in that case it's the 2/3 answer.

 

[Rolfe]

I've been hearing this puzzle for about 15 years, and I don't think I've ever heard it explicitly stated that the host is required to open a door. The puzzle is solvable only if that stipulation is added, and in that case it's the 2/3 answer.

Well, it's been a long time and I no longer have the original format. I do know that it was discussed as if he always offered a switch.

However, it's still, as II said, either 2/3 or 50/50, depending on Monty's intentions. It's the fact that you can't decrease your chance in that scenario that makes it "always switch", so that if he's deliberately avoiding the car you get the improved chance.

Rolfe.

 

[rastamonte]

In fact we would not know if the hustler knows which door the prize is behind. So all we can say is that by switching the probability of getting the prize will be from 50% - 66.7%. Determining the precise percentage will be a question of psychology.

When you say hustler, you must mean the three-card monte hustler. If so, you are wrong. He DOES know which card is the winning card. And he only offers you a switch if you have already chosen the winning card.

 

[Interesting Ian]

When you say hustler, you must mean the three-card monte hustler. If so, you are wrong. He DOES know which card is the winning card. And he only offers you a switch if you have already chosen the winning card.

I've already admitted my mistake. Read my post above

 

[Number Six]

No, it's always best to switch because although it might only be 50/50 chance whether you switch or not, the point is you don't know that. The chance of winning the prize is anything from 50% to 67% if you switch. Since you cannot determine the precise percentage, it is wise to switch.

If you don't know the mechanism Monty used to decide to open a door with a goat then you have no more information about whether the door you chose was the correct door than you did when you intially chose it.

If you know beforehand that Monty is going to reveal a door with a goat to you then it's best to switch. Otherwise, you can't know whether it's best to switch unless you can read Monty's mind.

 

[Number Six]

If he offers the switch then this makes it more likely as a general policy that either he always offers the switch or more often offers the switch than otherwise.

If you don't know the mechanism Monty uses to decide whether to reveal a door with a goat to you then you don't know whether his offer to switch makes it more likely that it's a general policy.

 

[Walter Wayne]

Well, it's been a long time and I no longer have the original format. I do know that it was discussed as if he always offered a switch.

However, it's still, as II said, either 2/3 or 50/50, depending on Monty's intentions. It's the fact that you can't decrease your chance in that scenario that makes it "always switch", so that if he's deliberately avoiding the car you get the improved chance.

Rolfe.

Yes you can decrease your chances by switching depending on Monty's knowledge and intention.

Lets suppose Monty intentions are to prevent you from winning the car. The simplest way for him to do this would be to accept your initial guess if you are wrong, and give the option to switch if you guessed right. This would mean that if he reveals a goat and gives you the option to switch, you have picked the correct door, and have a 100% chance to win if you stick with your guess, and 0% if you change.

More generally let us suppose that Monty's decision to reveal a goat is dependant on whether you chose right or not. Let's say:
p_c represents the chance Monty reveals a goat if you choose correctly first time
p_i represents the chance Monty reveals a goat if you choose incorrectly first time

Then the chance that you choose correctly and Monty reveals a goat is:
P(correct & goat) = p_c/3
Then the chance that you choose incorrectly and Monty reveals a goat is:
P(incorrect & goat) = 2p_i/3

Thus the odds of him revealling a goat are
P(goat)=P(correct & goat)+P(incorrect & goat)=p_c/3+2p_i/3

And then the chance of your initial guess being incorrect given that he reveals a goat is
P(incorrect given goat) = P(incorrect & goat)/P(goat)
P(incorrect given goat) = (2p_i/3)/(p_c/3+2p_i/3)
P(incorrect given goat) = 2p_i/(p_c+2p_i)

If the chance of your initial guess being incorrect given that he reveals a goat is greater than 0.5 you should switch.

Walt

 

[Rolfe]

Yes you can decrease your chances by switching depending on Monty's knowledge and intention....

Yes, yes, yes, I acknowledged that above. Maybe I worded it badly, but that post was predicated on Monty always opening one of the other two doors, even if you've originally guessed wrong.

Curt C said on that case it was a 2/3 advantage to switch, and I commented that this was only the case if he always deliberately revealed a goat - if he's opening either of the two remaining doors at random and sometimes reveals the car, there's no advantage.

Now, if you were trying to tell me how switching would decrease your chance of winning in a scenario where Monty ALWAYS opens one of the two doors you didn't choose and offers you the chance to switch to the third door, then I think you're going to have to run me through it again....

It's getting late....

Rolfe.

 

[Walter Wayne]

I see a misinterpreted the term "deliberately avoiding the car". I thought avoiding given in away as opposed to avoiding revealing it.

Damn it, I made a long winded post for nothing.

Walt

 

[plindboe]

Wow, this simply puzzle has surely been explained to death. I vote for Batman Jr. as the tutor offering the best explanations.