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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)

#321

Post by elteto »

lamontcg wrote: post #167

The asymmetry is that if you pick the one with the car first (only 1/3rd chance) then Monty Hall can show you either of the other doors. If you pick one without a car first (2/3rd chance) then Monty Hall must show you the door that doesn't have a car and the remaining door always has the car. So 2/3rd of the time you have a 100% chance of a car if you switch, 1/3rd of the time you have 0% chance of a car if you switc…

Fantastic answer. I went through and built the tree and it just shows everything so neatly. There's just a small mistake, when you say:

    4.  then have the contestant either switch or not switch and cut all your probabilities in half again.
You don't halve probabilities again because the strategy we are evaluating is _to switch_. So we always switch. It is deterministic.

If we switch deterministically we end up with 6 cases (leaves in the tree) with probability 1/9 where we win. 6/9 = 2/3!

If we switch randomly then we do split probabilities again, and end up with 6 cases with probability 1/18. 6/18 = 1/3.

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

#322

Post by digitalscribe »

hulium wrote: post #288
dekhn wrote: post #112

Earlier quoted context omitted.

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…

> opens another door, say #3, which has a goat There is no randomness in this sentence. The door has a goat, not a car. Then again, you might read it as an example outcome. But even if the door was chosen randomly, it would not justify the 50%/50% answer. Independently of that, assuming that the host chooses the door with the car and the goat with 50% probability each is the same mistake that confused so many, you ca…

This was an interesting discussion when I read about it long ago.

I don't think it's obvious that Marilyn's interpretation is the correct one. Two possible interpretations could be equally valid.

In law we have the "rule of the last antecedent" that says descriptive clauses modify the nearest antecedent noun.

Under this interpretation "which has a goat" modifies the exemplary door "say #3". Same as saying host opens a door, for example door #3 containing a goat. It could have been say door #2 containing a car.

Marilyn's interpretation is that "which has a goat" modifies "host opens another door." Same as saying the door opened by the host always has a goat.

Language is inherently ambiguous. I don't think either interpretation is unreasonable...

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

#323

Post by lamontcg »

elteto wrote: post #321
lamontcg wrote: post #167

The asymmetry is that if you pick the one with the car first (only 1/3rd chance) then Monty Hall can show you either of the other doors. If you pick one without a car first (2/3rd chance) then Monty Hall must show you the door that doesn't have a car and the remaining door always has the car. So 2/3rd of the time you have a 100% chance of a car if you switch, 1/3rd of the time you have 0% chance of a car if you switc…

Fantastic answer. I went through and built the tree and it just shows everything so neatly. There's just a small mistake, when you say: 4. then have the contestant either switch or not switch and cut all your probabilities in half again. You don't halve probabilities again because the strategy we are evaluating is _to switch_. So we always switch. It is deterministic. If we switch deterministically we end up with 6 c…

I don't really consider that a mistake it lets you compare the switching strategy to the not-switching strategy. YMMV.

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

#324

Post by elteto »

lamontcg wrote: post #323
elteto wrote: post #321

Earlier quoted context omitted.

Fantastic answer. I went through and built the tree and it just shows everything so neatly. There's just a small mistake, when you say: 4. then have the contestant either switch or not switch and cut all your probabilities in half again. You don't halve probabilities again because the strategy we are evaluating is _to switch_. So we always switch. It is deterministic. If we switch deterministically we end up with 6 c…

I don't really consider that a mistake it lets you compare the switching strategy to the not-switching strategy. YMMV.

But you wouldn’t be comparing them as much as evaluating a third strategy which is “randomly switch or not”. Right?

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

#325

Post by [deleted] »

bemusedthrow75 wrote: post #181
klyrs wrote: post #176

Earlier quoted context omitted.

A nice goat, you say? How much for a mean goat, jaded from years of being the spoiler prize that nobody wants?

Whenever someone talks about mean goats I think of only one thing: Buttermilk the baby goat, being a jerk. https://www.youtube.com/watch?v=AWvefaN8USk Enjoy.

[deleted]

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

#326

Post by lamontcg »

elteto wrote: post #324
lamontcg wrote: post #323

Earlier quoted context omitted.

I don't really consider that a mistake it lets you compare the switching strategy to the not-switching strategy. YMMV.

But you wouldn’t be comparing them as much as evaluating a third strategy which is “randomly switch or not”. Right?

I think I added them up separately and normalized by 50% because my brain loves to make shit hard.

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

#327

Post by EGPRC »

hgomersall wrote: post #306
tsimionescu wrote: post #278

Earlier quoted context omitted.

> the fact they all contain goats is pretty suggestive that I have the car, or at least 50/50 That's exactly the point. If he doesn't know, then it's exactly 50/50 and there is no reason to switch. If he does know, then it's 1/NUM_DOORS versus NUM_DOORS-1/NUM_DOORS, so you'd be crazy not to switch. The point is, if he picks at random, in the 100 door case, the vast majority of the time he will open the car door while…

This is why one can't ignore the car picked cases, so the simulator is correct.

I believe the confusion with "ignoring the car picked cases" comes due to thinking that they are additional games to the original ones, instead of being part of those that constitute the 2/3 in which the player starts picking wrong.

So you thought that eliminating them you were left with the original 1/3 vs 2/3, when you actually removed half of the 2/3.

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

#328

Post by WarOnPrivacy »

swayvil wrote: post #312
WarOnPrivacy wrote: post #57

Earlier quoted context omitted.

> The "smartest person in the world" tag is a marketing push. It reflects some broad understandings of the 1980s and what we valued.

She wears a suit with shoulder pads. Dates Michael Douglas.

Again, it was the 1980s. That's how we rolled. Also, Falling Down.

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

#329

Post by heleninboodler »

listenallyall wrote: post #301
heleninboodler wrote: post #292

Earlier quoted context omitted.

It's a lot easier to see the 2/3 nature if you imagine that instead of revealing the goat and then asking if you want to switch, Monty said "you've chosen this door that has a 1/3 chance. I'm offering you to switch to choosing both of these two remaining doors, at least one of which definitely has a goat by definition." This is functionally the same offer. It's 2/3 to switch because you're effectively choosing two do…

The initial door reveal is misdirection by an informed host, such that it has no effect on the odds, which are always, players choice of door is 1/3. Imagine instead: After the player picks a door, the host states out loud, "we started with 2 zonks, you picked one door, so there must be at least one unchosen zonk... and I'm going to show you one behind door #X (door opens with zonk)... now, would you like to switch?"…

> The initial door reveal is misdirection by an informed host, such that it has no effect on the odds, which are always, players choice of door is 1/3

Yes, exactly. Which means when you switch, the host is effectively allowing you to choose both the other doors, which gives you 2/3 odds.

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

#330

Post by lucas_membrane »

grogers wrote: post #285
harshreality wrote: post #249

Earlier quoted context omitted.

If you're playing the game as stated, and if the host opens a door revealing a goat, you must switch for the best odds. The offer to switch doors is part of the problem statement. You can't justify your decision based on the possibility in the real game that you might not get the offer to switch (which would mean that when you do it might be an intentional red herring). Even if there's a 50% chance there are all goat…

The person you are responding to is pointing out that if the host doesn't offer you to switch all the time, then it isn't the best strategy to always switch when you are given the option. For example, knowing that everyone switches when given the option, the host might only offer to switch when you've picked the correct door. In that case, not switching would be better. Hence the poker aspect.

I'll admit that it might be a little more complicated than that. Does life require that we make default assumptions? Is the default assumption that when we are seeing and playing a game for the first time, we should assume that the other players are following the simplest and most impartial strategy that can explain their behavior a good default assumption? Implicit default assumptions are rampant in human efforts to resolve the hard problems of life. Can anyone recommend ways to identify them, to find alternative default assumptions, choose good ones, and to be aware of their impact on conclusions.
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