Monty Hall

[andrewducker]

[Poll #1770413]

Explanation

I have known what the answer was for ages, but for some reason it only "clicked" in my head today. You can blame sarahs_muse for triggering it.

Comments

[andrewducker]

If she'd checked raptor two then that would leave universes 1 and 3. Which still gives me a 50% chance. Which one she checked doesn't matter.

[theweaselking.livejournal.com]

She's checked one of them.

In the set of all two-raptor pairs, you don't actually know which one she's checked.

My wording "the first one" really should have been the more clear "I have checked one". The traditional Science Announcement is "Yes, at least one is female".

All you know is that they're not both male.

[andrewducker]

That's not true.

I know either that I'm in universes 1/3 or in universe 3/4. In either case I have a 50% chance of the other one being male and the other one being female.

[theweaselking.livejournal.com]

Only if you accept "I've checked one and it's female" to mean "I have numbered them, and determined that #1 is female, thus eliminating TWO of the possible results", instead of the intended (and sloppily worded) "I have determined that one of them is female, without caring which"

[andrewducker]

It depends.

Let's assume that there are two doors, each with a raptor behind it.

If they say
"I went behind door 2 and checked, and that one is a female." then we've eliminated two of the results.

If they say "I went behind both doors and checked, and I remember that at least one of them was a female, but I can't remember which one." then we've only eliminated one of the results.

I think this question requires more careful phrasing than the Monty Hall one :->

[chuma.livejournal.com]

"If they say "I went behind both doors and checked, and I remember that at least one of them was a female, but I can't remember which one." then we've only eliminated one of the results."

This example would indeed be a 33% chance of the 3 remaining options.

[theweaselking.livejournal.com]

It does.

But I love this kind of problem because of the *breadth* of the wrong answers you get, and the paths people use to get there.

(I just, here at work, got two more smart educated geeky people to fall for both the Monty Hall and the puppies.)

[chuma.livejournal.com]

I'm sorry but the only wrong answers here are your own. If I could drag others into this argument, I know a statistician who I am sure could explain it more succinctly than I could.

[theweaselking.livejournal.com]

Au contraire, there are at least three wrong answers in the original poll, and at least two wrong answers to the beagle puppies problem. Not counting the two in my office just now!

This is not to say that my answers to this kind of thing are always correct - the Bob And Sue are rolling dice one had me bouncing off wrong answers for hours before I found the trick - but I really do love this kind of problem and this kind of thread because I love seeing *how* people come up with the answers they reach.

[andrewducker]

Yes. Seeing how people's intuitions are broken is always fascinating. Especially when they're mine!

[theweaselking.livejournal.com]

And now you have another example, of mine getting all twisted up!

[atreic.livejournal.com]

I understand this now! You're wrong, because 'I have checked one and it is female' is not the same as 'at least one is female'. Some cases of the latter will not be picked up by the test.

Ie, if the raptor family is a boy and a girl, it would always be included in 'at least one is female'. But half the time it would not be included in 'I checked 1 and it was female' as you might have checked the boy.

Or as a friend of mine said so much better than I could '"First-born" and "first one I looked at" are equivalent as far as probabilities are concerned, and the probability is 1/2.

If instead I took a sample of both velociraptor's DNA, mixed them together and said "Hey I've found some Y chromosomes, at least one velociraptor is male" THEN the probability that both are male is 1/3.'

[lizw.livejournal.com]

If instead I took a sample of both velociraptor's DNA, mixed them together and said "Hey I've found some Y chromosomes, at least one velociraptor is male" THEN the probability that both are male is 1/3.'

This is brilliant.

[damerell.livejournal.com]

"I have checked one and it is female" obviously is not the same as "at least one is female". The latter statement may imply having checked both.

[theweaselking.livejournal.com]

The intended interpretation was that the owner of the park wouldn't know which was which - similar to "One is female, the other is OH GOD IT HAS MY NECK ARGH WHY DID WE BREED RAPTORS WHYYYYYYY"

[del-c.livejournal.com]

The owner knows she checked one and stopped. The stopping is critical, because it rules out one of the two ways to find only one female (the way ruled out is to find a male, say "that sucks", and then find a female).

The 2:1 odds (1/3 probability) solution requires there to be two ways to get one female, against only one way to get two females. Here, there's one way to get one female, and one way to get two females, and that's all. So the solution is 1:1 odds (1/2 probability).

By killing the scientist, the deadly raptor stops her playing word games with the park owner; otherwise she would have mischievously turned the uniquely distinguishable states "MF" and "FM" into the two indistinguishable states "one F". But these aren't just word games. Our measurements of chemical entropy confirm that the loose verbal description "this is a gas: the molecules are all over the room" really does describe many more possibilities than "this is a solid: the molecules are in this dish" does.