Monty Hall Problem

[Gravy]

With this puzzle you've been shown 59 cars and 39 goats, right?

If I'm reading that right, you had a 60% chance of guessing right to begin with, so you retain that percentage by staying. Switching would give you a 40% chance of getting the car since all probabilities have been collapsed to those two doors.

Of course, this assumes the host deliberately chose to leave one car and one goat uncovered.

What do you think is the probability of getting a prompt response from the person who made that post four years ago?

 

[IMST]

What do you think is the probability of getting a prompt response from the person who made that post four years ago?

Wow.

Saw an active thread, never occurred to me to see if the last page contained posts from '04.

Always amazing to see these threads brought back from the dead like that.

ETA: Art Vandelay's last post: 08/07. I'ma change doors please Monty.

 

[lionking]

I just tried this problem on my family, and they all gave the intuitive 50/50 answer. So I used the three cards, and most still didn't - except my wife who would acknowledge that she is the least (formally) educated of the lot. She saw it immediately.

 

[Rolfe]

Once upon a time (perhaps in the recesses of this thread), I thought I had my brain around this.

Now I may be remembering wrong, but I thought the conclusion I came to was, switch, whatever, given the problem as set out.

A. If Monty knows where the car is and is deliberately avoiding it, then you improve your chances.

B. If Monty is as ignorant as you are, then the chances stay at 50:50.

If you have no idea which scenario Monty is running it makes sense to switch. Because B is the worst case scenario (unless the game is truly bent), and in that one, while you don't gain by switching, you don't lose either.

By sticking, you are rejecting the possibility of improving your chances if A is in fact the scenario, and getting no possible benefit out of it.

By switching, you allow yourself to take advantage of the possibility of scenario A being the situation, without exposing yourself to a decrease in your chances.

Thus, if the rules are not specified, and A is within the bounds of possibility, then by definition switching is the sensible thing to do.

Now I'm waiting for everyone to point out the three-year-old post where someone proved me wrong about this.

Rolfe.

PS. Billydkid, I approached it thinking about 100 doors and one car. A million does rather seem like overkill.

 

[TheBoyPaj]

You're right. You should always switch.

 

[GodMark2]

Once upon a time (perhaps in the recesses of this thread), I thought I had my brain around this.

Now I may be remembering wrong, but I thought the conclusion I came to was, switch, whatever, given the problem as set out.

A. If Monty knows where the car is and is deliberately avoiding it, then you improve your chances.

B. If Monty is as ignorant as you are, then the chances stay at 50:50.

If you have no idea which scenario Monty is running it makes sense to switch. Because B is the worst case scenario (unless the game is truly bent), and in that one, while you don't gain by switching, you don't lose either.

By sticking, you are rejecting the possibility of improving your chances if A is in fact the scenario, and getting no possible benefit out of it.

By switching, you allow yourself to take advantage of the possibility of scenario A being the situation, without exposing yourself to a decrease in your chances.

Thus, if the rules are not specified, and A is within the bounds of possibility, then by definition switching is the sensible thing to do.

Now I'm waiting for everyone to point out the three-year-old post where someone proved me wrong about this.

Rolfe.

PS. Billydkid, I approached it thinking about 100 doors and one car. A million does rather seem like overkill.

C: Monty may be evil, and thus only open a second door if he already knows you picked the car first, but not offering you the choice when you pick a goat.

Include that possibility in the mix, and switching (when offered) is not necessarily the best choice. With only a single instance from which to determine Monty's motive and the rules of the game, you don't have enough information.

 

[GreyICE]

Once upon a time (perhaps in the recesses of this thread), I thought I had my brain around this.

Now I may be remembering wrong, but I thought the conclusion I came to was, switch, whatever, given the problem as set out.

A. If Monty knows where the car is and is deliberately avoiding it, then you improve your chances.

B. If Monty is as ignorant as you are, then the chances stay at 50:50.

If you have no idea which scenario Monty is running it makes sense to switch. Because B is the worst case scenario (unless the game is truly bent), and in that one, while you don't gain by switching, you don't lose either.

By sticking, you are rejecting the possibility of improving your chances if A is in fact the scenario, and getting no possible benefit out of it.

By switching, you allow yourself to take advantage of the possibility of scenario A being the situation, without exposing yourself to a decrease in your chances.

Thus, if the rules are not specified, and A is within the bounds of possibility, then by definition switching is the sensible thing to do.

Now I'm waiting for everyone to point out the three-year-old post where someone proved me wrong about this.

Rolfe.

PS. Billydkid, I approached it thinking about 100 doors and one car. A million does rather seem like overkill.

You ignore the fact that A and B are the same scenario. EXACTLY the same. If he opens it at random and happens to hit the goat, its exactly the same if he knew he hit the goat.

If he opens it at random and hits the car, you ought to know your chances of getting the car. DUH

 

[Rolfe]

C: Monty may be evil, and thus only open a second door if he already knows you picked the car first, but not offering you the choice when you pick a goat.

Include that possibility in the mix, and switching (when offered) is not necessarily the best choice. With only a single instance from which to determine Monty's motive and the rules of the game, you don't have enough information.

Yes, I remember that possibility being discussed earlier. I think I was including that in the category of "bent" games, though I accept that's debatable.

Rolfe.

 

[Rolfe]

You ignore the fact that A and B are the same scenario. EXACTLY the same. If he opens it at random and happens to hit the goat, its exactly the same if he knew he hit the goat.

If he opens it at random and hits the car, you ought to know your chances of getting the car. DUH

I'm going to bed now. Somebody else can have a pop at this one.

Rolfe.

Hint: Think about the hundred doors and one car, and whether or not you'd switch if the choice were still on offer when you were down to only two doors left shut - your door and one other.

 

[GreyICE]

I'm going to bed now. Somebody else can have a pop at this one.

Rolfe.

Hint: Think about the hundred doors and one car, and whether or not you'd switch if the choice were still on offer when you were down to only two doors left shut - your door and one other.

Rolfe, you're not getting it. Obviously you switch. But your scenario B isn't a scenario. It doesn't matter whether Monty opened the door at random or with foreknowledge, as long as he hit a goat.

He reveals the same information either way. It's your lack of understanding that makes you think that they're different scenarios.

The only way it doesn't influence the outcome is if the event is 50:50 going in, and you lose if it hits the car.

The interesting aspect of the Monty Haul problem is that you can use the outcome of one completely random event to influence the outcome of another event.

It's the same thing with the two kids problem - the knowledge of one random event gives you better odds on guessing the outcome of the other random event, despite the fact the two are not linked.

 

[ravdin]

I probably don't have anything new to add to this discussion. This came up with a group of my coworkers several years ago. We argued about it for a long time and I came up with the best arguments I could- (e.g. 100 doors, walking them through NOT switching several times to show that it really does only work for them 1 out of 3 times).

My friend then came up with the best argument I've heard yet, which was this: if you still don't believe that you should switch, meet us in the alley after work for a special game. Bring a roll of 20 dollar bills with you.

By the way, if you're not convinced by the 100 door argument, I'll play that game with you any time you like. I'll even give you 50 to 1 odds in your favor!

 

[gnome]

The ambiguity is in the problem statement, that doesn't indicate whether monty will show you a goat after your choice every time. If he will, then switching is a great idea. If he might not, then it isn't.

If you want the problem to have it's classic answer, you must phrase it that he would always show you a goat.

 

[CurtC]

You ignore the fact that A and B are the same scenario. EXACTLY the same. If he opens it at random and happens to hit the goat, its exactly the same if he knew he hit the goat.

No, if he is opening a door randomly, then your chances aren't improved by switching (it's a 50/50 choice at that point). The "2/3 by switching" answer works only if the host knows where the goat is and purposely avoids it.

 

[jsfisher]

No, if he is opening a door randomly, then your chances aren't improved by switching (it's a 50/50 choice at that point). The "2/3 by switching" answer works only if the host knows where the goat is and purposely avoids it.

Exactly right.

1/3 of the time, Monty choosing at random will reveal the car. Game over.
1/3 of the time, Monty will reveal a goat and your door has a goat.
1/3 of the time, Monty will reveal a goat and your door has a car.

50/50 with random Monty switching will help.

 

[GreyICE]

No, if he is opening a door randomly, then your chances aren't improved by switching (it's a 50/50 choice at that point). The "2/3 by switching" answer works only if the host knows where the goat is and purposely avoids it.

No it's not. As soon as he opens a door with a goat behind it, you gain certain information about your choice. If he opens a door with a car behind it, you also gain information about your choice.

While the randomness does not increase your chances of winning at the OUTSET of the game, in the middle of the game, if a door with a goat is opened, it behooves you to switch.

The interesting part of the Monty Haul problem isn't that it increases your chances if he opens a door with a goat - it's that you can obtain information about unknown members of a random system given information about other members of the system even though every bit of information was obtained randomly. Meaning despite the fact that Monty opens the door randomly, once he's opened the door, the information you received from the door opening randomly informs you about other parts of the random system. This means that you can obtain information even if there's no Monty - i.e. no one knows how the system works.

This becomes more interesting with the baby problem (where the outcome is less deterministic and seemingly more random). There you can also see how the game works with no Monty - by gaining the information that one child is a son, you learn statistical chances about the other child's gender despite the fact that no one knows how gender is determined.

 

[Herzblut]

No, if he is opening a door randomly, then your chances aren't improved by switching (it's a 50/50 choice at that point). The "2/3 by switching" answer works only if the host knows where the goat is and purposely avoids it.

This is impossible. Synaptic structures in his brain have absolutely nothing to do with your winning chances in this phase of game. I mean, by what kind of magic shall there be any kind of relationship?

The winning chance of your initial choice is still 1/3 and the probability that the Ferrari is behind one of the two other doors is still 2/3. That's what counts.

 

[GreyICE]

This is impossible. Synaptic structures in his brain have absolutely nothing to do with your winning chances in this phase of game. I mean, by what kind of magic shall there be any kind of relationship?

The winning chance of your initial choice is still 1/3 and the probability that the Ferrari is behind one of the two other doors is still 2/3. That's what counts.

Finally!

Thanks.

This has some amazingly counter intuitive implications.

 

[Herzblut]

This has some amazingly counter intuitive implications.

What you mean? The daughter/son questions?

 

[GreyICE]

What you mean? The daughter/son questions?

Yeah, not the monty haul one (since now I'm confused if I'm remembering correctly, I just charted it out, and I remember that the odds increase might be from the behavioral change, since it's refusing to show up again. Also, I hate that problem. Did I mention that I hate that problem?).

Edit: No, done charting. It's conclusively 50/50 if the door is opened at random. Even if you learn the outcome. It's only if the host is benevolent. Goddamn it Monty Haul. Even if you don't care if the host opened the door at random, it turns out the universe does.

 

[Vorpal]

The following are probabilities of winning the game.
(1) Monty knows where the car is

Reveals random goat|Keep*: 1/2|Switch: 1/2
Goat, not the player's|Keep: 1/3|Switch: 2/3

(2) Monty doesn't know where the car is, and Monty revealing the car counts as an automatic win:

Reveals random door|Keep: 5/9|Switch: 2/3
Not the player's|Keep: 2/3|Switch: 2/3

(3) Monty doesn't know, and Monty revealing the car counts as an automatic loss:

Reveals random door|Keep: 2/9|Switch: 1/3
Not the player's|Keep: 1/3|Switch: 1/3

P.S. The disagreement with CurtC and jsfisher is apparently because they're talking about different conditions than those who object to them. If Monty (or the player, for fairness) picks a door to be revealed at random that is not the player's first guess, then they are correct in that keeping or switching doesn't matter. However, it is interesting that in all cases, switching does no worse than keeping.

Edit: Clarification: (*) this 1/2 probability switches if Monty's pick was the player's pick, otherwise keeps. If keep regardless of outcome, it is 1/3, although this is a very silly thing to do.