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Marilyn vos Savant and the Monty Hall Problem (2015)

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Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#121

Post by kristjansson »

dekhn wrote: post #93

There's the Monty Hall Problem and then there's the Second Order Monty Hall Problem Problem. That is, did the wording as it was originally printed unambiguously define the problem to be solved, or was it ambiguous enough that some people who got it wrong got it wrong for the right reason? Like many others, I completely missed that when Monty opens a door to show you a goat, he always shows you a goat. I think under c…

As someone who misunderstood the problem as you describe, I'm much more invested in this version of the problem :)

The problem statement is just a mess:

> You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat.

even tiny edits like:

> You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door which has a goat, say #3.

or even

> You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which he knows has a goat.

make it clear that the last phrase is describing an invariant of the problem, not an artifact of the illustrative hypothetical.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#122

Post by dboreham »

Scarblac wrote: post #105
dekhn wrote: post #93

There's the Monty Hall Problem and then there's the Second Order Monty Hall Problem Problem. That is, did the wording as it was originally printed unambiguously define the problem to be solved, or was it ambiguous enough that some people who got it wrong got it wrong for the right reason? Like many others, I completely missed that when Monty opens a door to show you a goat, he always shows you a goat. I think under c…

I assume that at the time of writing, most readers would have been familiar with the actual game show and the fact that the host always opens a door with a goat?

I had never heard of the show when I was introduced to the problem so all my knowledge comes from the problem description.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#123

Post by jncfhnb »

bemusedthrow75 wrote: post #119
jncfhnb wrote: post #107

Earlier quoted context omitted.

If he only opens a door when you picked the car, then you will lose 100% of the time by switching when he shows a goat. I instead love when people get uppity about others being wrong about Monty hall while still being wrong about Monty hall!

It doesn't matter what he only does, or always does. It matters what he did in the problem as it is described. Because that is what you're solving.

The problem as described does not specify his behavior. Only what happened. If all you know is that he opened a goat door you have learned nothing.

It could be that he always opens a goat door meaning you should switch.

It could be that he only opens a goat door if you picked the car and you should not.

It could be he only opens a goat door if you picked a goat and you should.

Probability cannot determine which of these is any more likely than the other. You have learned nothing.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#124

Post by Jimmc414 »

grumbel wrote: post #61
Jimmc414 wrote: post #35

I never understood this one. Not choosing a new door seems like an implied 50/50 choice to keep your original selection. edit: I know it works out on paper, but if I put myself in the situation, I know that if I decided not to choose again that is a choice and I am re-choosing the same door.

The remaining doors are not random. The one you picked still has a 1/3 chance of winning, that doesn't change with opening another door. However switching doors is essentially the same as picking all the other doors at once, since the door with a goat is not opened, thus the 2/3. Imagine the thing with 100 doors and imagine the host isn't opening any doors, but just telling you what's behind them and it becomes prett…

>The one you picked still has a 1/3 chance of winning

If there are only two choices remaining, I guess I don't understand the assertion that there is still a 1/3 chance of winning.

Suppose you were new to the contest and were allowed to put money down on the choice at the point that there were two doors left.

Then you repeated the bet every new contestant.

Would one door would be more profitable than the other?

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#125

Post by JackFr »

dekhn wrote: post #112
ghaff wrote: post #101

Earlier quoted context omitted.

I think this is the original "Ask Marilyn": Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? Craig F. Whitaker Columbia…

Yes, I'm literally talking about this sentence: "host, who knows what’s behind the doors, opens another door, say #3, which has a goat." I parsed that as "50% of the time, monty opens another door and it has a car and you win immediately, and 50% of the time, monty opens another door and it has a goat". In retrospect I think my brain just sort of pictured that and proceeded to assume there was no reason to switch, an…

> 50% of the time, monty opens another door and it has a car and you win immediately

Why would you think you won? You didn't choose that one with the car and you aren't offered the opportunity to switch to the open one, only the closed one. The only interpretation which makes sense is that he always shows you a goat.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#126

Post by btilly »

saalweachter wrote: post #108
orblivion wrote: post #85

Earlier quoted context omitted.

I've never seen the show; I'm curious, did he always open one of the doors?

I mean, the original run ended and the first revival also went bankrupt before I was born, but my understanding of the show is that the host always presented the three doors [at the end, to the show's winner, similar to the prize puzzle at the end of Wheel of Fortune] and always eliminated one of the losers. Fun Facts which I think I'm remembering correctly: it wasn't always a goat, they would have other random anima…

So you made an assertion about what happened 100% of the time on the show, when you don't even remember it???

I went and re-read the interview with him that I'd read decades ago. It turns out that we're both wrong. The game was more complicated. He not only knew where the prize was, he was offering people money about whether or not to switch. He knew the answer, and was trying to manipulate the psychology of the person in front of him for the sake of entertainment. Whether he was on the side of the contestant depended on his mood. His quote about the actual game was:

*My only advice is, if you can get me to offer you $5,000 not to open the door, take the money and go home."

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#127

Post by full_monty »

niel wrote: post #86
tromp wrote: post #66

Earlier quoted context omitted.

Your formulation is once again ambiguous. Instead of clearly stating that Monty had to open 998 empty doors, you state that he did that. Which could mean he happened to do that. Had to vs happened to makes a big difference. Had he opened them at random and by (extremely small) chance they happened to be empty, the odds are quite different.

I'm trying to understand why the odds would be different if Monty opened empty doors by chance versus on purpose. Does it really change anything?

Yes.

If Monty opens the door by chance there are 3 equally likely cases: There's a 1/3 chance you picked the car and Monty shows you a goat, 1/3 chance you picked a goat and Monty shows you the other goat, 1/3 chance you picked a goat and Monty shows you a car. So if Monty shows you a goat you have equal probability of being in one of the first two cases.

If Monty doesn't open the door by chance then he never shows you a car. So 2/3 of the time you picked a goat and Monty shows you the other goat.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#128

Post by bemusedthrow75 »

jncfhnb wrote: post #123
bemusedthrow75 wrote: post #119

Earlier quoted context omitted.

It doesn't matter what he only does, or always does. It matters what he did in the problem as it is described. Because that is what you're solving.

The problem as described does not specify his behavior. Only what happened. If all you know is that he opened a goat door you have learned nothing. It could be that he always opens a goat door meaning you should switch. It could be that he only opens a goat door if you picked the car and you should not. It could be he only opens a goat door if you picked a goat and you should. Probability cannot determine which of th…

"If all you know is that he opened a goat door you have learned nothing."

This is precisely where you are wrong.

Re: Marilyn vos Savant and the Monty Hall Problem (2015)

#130

Post by jncfhnb »

bemusedthrow75 wrote: post #118
jncfhnb wrote: post #102

Earlier quoted context omitted.

Incorrect. The host doesn’t need to choose randomly. His purposeful behavior could also be random. The only way you know the switching to be optimal is if it specifically says he picks a goat door to open. If it says he opens a door, and it has a goat, then you’ve learned nothing about the remaining two.

> His purposeful behavior could also be random. It doesn't matter. Because the problem explains what happens: he opens a door and reveals a goat. That is crucial information. There's now only one goat and one car left. But the choices have not been shuffled: you're definitely still pointing at your original choice. What were the odds that your original choice is a goat? Two in three. If you picked a goat, what is gua…

Incorrect. You have no way to know if he would only show you a goat if you picked the car.

If that were the case then switching is a guaranteed loss. Simply knowing that he opened a door and showed a goat does not mean he would have done this regardless of your choice.

Not a probability problem without this info.

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