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

[chuma.livejournal.com]

Why should I accept only £1? You told me that the odds of HH were the same as HT and TH. That would mean you believe the odds of HH are 33% which is 2-1.

[theweaselking.livejournal.com]

Because *you* say it's 50/50.

And since you insist it's a fair bet, I will happily take your money as if it was a fair bet. Giving you 2-1 means we'll break even on average, and that doesn't make the point nearly as well as you going broke betting the farm that a 2/3 chance is 50/50.

[chuma.livejournal.com]

And because the scenario you described is not the same as the one in the question you gave. In the one you just offered you get to look at BOTH coins. That doesn't happen in the raptor question. What happens in that one is that a single raptor is looked at and its sex is determined. What you are doing is looking at both of the coins and letting me know that AT LEAST one of them is a Head. That is not the same thing at all and infact in THOSE circumstances the odds ARE 33% of it being 2 Heads.

You see the major difference now?!?

[bracknellexile.livejournal.com]

I think, after a long evening of posting, I see the problem.

It was clarified elsewhere in the threads that the original raptor question was worded sloppily and "the first one's female" was meant to imply, "the first one I looked at is female but it doesn't matter which one of the two raptors it was" - or more simply, "one of them is female".

I suspect, if you missed this clarification early on, Si (and I'm not sure if it was in a thread of the comments you were in or not and there's waaay too many to check now) that it would explain all the arguing.

Everyone step away from the interwebs and take a deep breath?

[chuma.livejournal.com]

Just had a look and found the "clarification" The below is taken from his conversion with Andy in a different branch of the thread.

"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."


I'm afraid even this is contradictory. If she checked only one then it would still be 50% (she can't be certain at least one is female without knowing that the only one she saw is female). If she saw both of them, that would change matters entirely (She could see the first was male and still see the second and use the same phrase were it a female). This was the difference between the coin scenarios above too.

This started as a thread on Stats, but sadly has come to be one on English and wording of questions. I'd be less annoyed if I hadn't wasted time on trying to explain myself to someone who had access to a whole webpage describing the differences between the two and the pitfalls of being ambiguous.

*sigh*

[theweaselking.livejournal.com]

I actually do see your argument now, yes.

I still think you're *wrong*, but now I understand where you're coming from

The problem here is that you're not supposed to know which coin you looked at - only that examination of the coins has shown that one is heads.

You, instead, are aiming to exclude all cases where the first coin examined is tails, which *would* result in 50/50 - except we're not discussing that. Not even in the original question, where I phrased it poorly and implied a possible ordering to the raptors.

Instead, the original phrasing, corrected and clarified three hours ago, was meant to imply that examination of a random raptor from the pair had resulted in a female, and the second raptor had not yet been examined.

Which, given our original four equally likely possibilities of
MM
MF
FM
FF
Means we're SOMEWHERE in the bottom three, but we don't know which.
We have either FX or XF, and in 2/3 of the possible cases X is M.

[chuma.livejournal.com]

ARGH!

Thats STILL 50%

The point is the 33% comes from being able to look at BOTH of them and say that at least one is female!

If you look at the first one and it is female you can say with certainty "at least one is a female" whether or not you look at the second, which happens 50% of the time. The point is that if you see the first one is male then you can STILL look at the second one and if that is female you can STILL say "at least one is a female", which happens 25% of the time.

In THAT scenario it is 33% that both are female. Only identifying ONE of them without looking at the other means that you CANNOT assume ANYTHING OTHER THAN the one she looked at IS female. That means the other one is the ONLY unknown and therefore it is a 50-50.

The key to phrasing this is that a female can be either or both rather than a specific one.

Please PLEASE try and understand this. Read your own link again and then maybe we can agree and move on.

[theweaselking.livejournal.com]

The problem here is the same one you had with Bob and Sue: We're talking about a single specific instance of a larger generalisable problem.

Just because we know Bob won this one game, doesn't mean Bob always wins games, so we can't discount Sue when determining *how often* Bob wins. Just because Sue *can* win games doesn't mean that she might have won this specific game, so we need to take into account that we know she didn't win this one.

Back to the babies: We haven't addressed what might have happened has the first baby been male, because in our specific, that didn't happen.

But we didn't generate raptor pairs guaranteeing one is female.

We generated a random pair of random raptors, and then learned that one, or the other, or both, was female.

The fact that Sue didn't win the game in question doesn't mean that Sue can't win, and thus that Sue's existence doesn't alter the distribution of Bob's wins.

The fact that the Head Scientist determined that one or both of the raptors was definitely female did not change the fact that they *could* have both been male.



You appear to be arguing that by examining only one of them, the head scientist has introduced an ordering to the raptors - the examined and the unexamined......

....and, I think I actually just got what you're aiming at.

Rephrase as "This one is female! The other one escapes and eats you. Was it male or female?" and suddenly the question becomes much clearer.

So you're right, and I do see your point. In the situation where you know which one she examined, it is not the same as "at least one is female".

From the perspective of andrewducker, who owns Jurassic Park 3.0, though, he only knows that the scientist knows one is female without knowing which. If she was to tell him "one is female, argh they've got my neck blergh I am teh dedzorz" all Lovecraft-character-blogging-his-execution-style, though, wouldn't the question of "were both the raptors who ate the scientist female" sit on 33%?

[chuma.livejournal.com]

Deleted my comment as I went off on one without reading the second half of your comment, which I apologise for.

Yes. You got it. By rephrasing it as you have, it is now 33%. Agreed.

[chuma.livejournal.com]

Also with regards to Sue and Bob, I think a better way to phrase it is "Of the times that Bob wins, what proportion will be on his 2nd turn?", which would rule out any ambiguity. I appreciate that this was not YOUR fault as the question was taken as a whole from another forum, but it is misleading none the less.