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

[theweaselking.livejournal.com]

More particularly: There's no way to tell if the first one she checked was in either position. So all you know is that they're not both male.

[chuma.livejournal.com]

It's irrelevent. If you are given a fact that one is female, and assuming there are no hidden facts like eggs from the same litter have to be the same gender, then it is a 50-50 that the other one is female.

[theweaselking.livejournal.com]

Nope! The trick is that you're not determining the sex of a random baby, you're determining the sex of a random baby from a set of two random babies.

The possible sets of two random babies are:

MM
MF
FM
FF

All of those two-baby sets are equally likely.
You know that one of the two babies is female, but not which one of the two - your Head Scientist's slightly misleading "first one" is "the first one she checked" and, if the original comment could be edited, it would be by now.

But!

All you know is that you have four equally-likely possibilities... and you're DEFINITELY not in possibliity #1, "MM".

This leaves you with three possible combinations:

MF
FM
FF
where are least one baby is female.

And in 2/3 of the cases, the second baby is male.

(This is a classic nonintuitive result, like Monty Hall's problem.)

[chuma.livejournal.com]

Wrong.

You are not weighting your chances. Your options are MF FM or FF but the chance of having a male are 50-50. You then have a 50% chance of the Male being in place 1 or 2. Thus you have 25% chance of MF and FM and a 50% chance of FF.

[theweaselking.livejournal.com]

Intriguing argument.

Imagine that Monty Hall has three goats and a car, behind four doors. The car is equally likely to be behind each door, such that your odds are:

1: 25%
2: 25%
3: 25%
4: 25%

Monty opens door #1 to reveal no car.

*I* say that your odds are now
1: 0%
2: 33%
3: 33%
4: 33%

*You* say that your odds are now:
1: 0%
2: 25%
3: 25%
4: 50%

I am wondering why you think this is.

[chuma.livejournal.com]

You chose a door before you were shown a goat in one of the other 3. The host KNOWS that that door didn't have the car behind it before offering you a swap. He revealed extra information about the system of 3 doors after chosen. The important distinction is that this information was not a given at the point you chose the door as that the host knows for certain one of those doors was with a goat inside.

[theweaselking.livejournal.com]

You chose a door before you were shown a goat in one of the other 3.

Very true, and the fact of that choice was why the odds acted in a nonintuive way.

In this problem, he's doing the opposite. He's *not* adding information by telling you that the car is not behind door #1, so the odds of the car being behind the other three doors should be equal, right?

[chuma.livejournal.com]

Although if you wish to apply the above to the solution, you need to add % chances :-

Before you know one is female the options are:
MM 25%
MF 25%
FM 25%
FF 25%

AFTER you know one is female the options are:
MM 0%
MF 25%
FM 25%
FF 50%

[theweaselking.livejournal.com]

Why does the lack of MM make the MF possibilities less likely?

[chuma.livejournal.com]

It doesn't. There is a 50% chance of FF or MF/FM. The location of Male or Female doesn't change the fact that each option is 50% as a whole. There is a 50-50 that the Male is in the 1st or 2nd slot so that is half of the 50% (25%) for each option.

Your method is not a Monty Hall problem. That is when you are given NEW information AFTER a decision has been made. This example gives informations BEFORE a decision has been made.

[bracknellexile.livejournal.com]

Nope. you want to know if you have two females.

Your scientist chooses one and checks it. You have information. You now want to know the odds of the other being female. That's a re-evaluation of the probability after you've been given information.

[chuma.livejournal.com]

That doesn't seperate out the system.

Let me apply this to dice. I roll 2 dice. I reveal that one of them is a 6 but don't reveal which dice. The options are 1-6 2-6 3-6 4-6 5-6 6-1 6-2 6-3 6-4 6-5 6-6. You will notice that 6-6 only appears once in the list. Do you honestly believe that I have less chance of rolling a 12 than rolling a 7?

[bracknellexile.livejournal.com]

Without a position, i.e. the, "I've checked one and it's female" as discussed elsewhere, surely that's:

MM 0%
MF 33%
FM 33%
FF 33%

[chuma.livejournal.com]

Okay if we are going to stick to this modal, Lets assume it is a FF scenario. How do you know the first female she saw is the one in the 1st slot and not in the 2nd? If you want to keep this option then the true options are:

M F*
F* M
F* F
F F*

Where the F* is the one that we know about. Now given them all 25% and you have a true modal.

[theweaselking.livejournal.com]

So you think the FF probability really does double once you learn that either of them is F?

[bracknellexile.livejournal.com]

But that's not the (revised) wording. In your scenario you're saying, "I've checked -that- one and she's female." You've added position to it.

[chuma.livejournal.com]

If you are applying it to a position, why do you have MF and FM? You can't have it both ways!

[atreic.livejournal.com]

In the interest of supporting people who appear to be right on the internet, I've read the wikipedia page and thought about it, and you appear to be right.

The crux is clearly

From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.

From all families with two children, one child is selected at random, and the sex of that child is specified. This would yield an answer of 1/2.

You (or nature, or hot velociraptor sex, or the scientist breeding velociraptors, or people who don't understand stats on the internet) have selected a family of two velociraptors at random from the set of all possible families of two velociraptors, and have then acquired the additional information that one velociraptor is a girl. It's clearly the second scenario, not the first. Unless the scientist was producing embryos in some strange probability space where they had to have at least one girl embryo. But if they'd done that they wouldn't need to check them, because they'd know they had at least one girl ;-)

[chuma.livejournal.com]

Bless you :)

I knew someone would put it more succinctly than me. :D

[atreic.livejournal.com]

[In the interests of transparency I think I have to add that I am still managing to confuse myself every time I think about this.]

[atreic.livejournal.com]

I have made an LJ poll to confuse myself further, and friended you so you can read it if you want to and are not entirely sick of this debate already ;-)

[ptc24.livejournal.com]

My reading of "the first one" means the options are:

MM 0%
MF 0%
FM 50%
FF 50%

[bracknellexile.livejournal.com]

Doesn't "the first" imply a position?

If she'd said, "I've checked one, and it's female," rather than "I've checked the first one, and it's female," then I'd agree with you. I can see your point, but the implied position in the quote means I'm with Andy on this one... for now :)

[theweaselking.livejournal.com]

You're right, it's a bad phrasing. "the first one" was meant to mean "I have checked one, and will soon check the other".

Either way, there's no position.

[bracknellexile.livejournal.com]

Yeah, I saw your reply to Andy below. With the revised phrasing and no position, I agree that it's a 2/3 chance the other is male.