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GAINING WEIGHT...
Mar 26, 2007

See? Science proves the JewsMuslims are inferior and must be purged! I'm not a racist, honest!

Phyzzle posted:

Bob has two children. The younger one is a girl. What are the odds that the older one is a boy?

About 50%

Bill has two children. One is a girl. What are the odds that the other one is a boy?

67%. Yes.

Explain yourself

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GAINING WEIGHT...
Mar 26, 2007

See? Science proves the JewsMuslims are inferior and must be purged! I'm not a racist, honest!

Phyzzle posted:

There are 4 equally likely outcomes when you have 2 kids:
Boy Boy
Boy Girl
Girl Boy
Girl Girl

In Bob's case, we know
B G
G G
Are the only two possibilities.

In Bill's case,
B G
G B
G G
Are three equally likely possibilities.

I'm not sure about this. I think throwing in order of birth is just a red herring. B G and G B are the same "possibility" for Bill; each family can either have two girls or one of each. Fifty fifty.

Take it to the extreme; a family has 100 kids and 99 of them are girls, but we don't know the order. Does that make it a 99% chance the other is a boy? No. Each birth is a 50/50 probability.

computer parts posted:

Sounds like a good time to introduce the Monty Hall problem.

This isn't quite the same as Monty Hall, where the fact that there are three distinct options is beyond question, and the fact that we know there IS a goat SOMEwhere.

GAINING WEIGHT...
Mar 26, 2007

See? Science proves the JewsMuslims are inferior and must be purged! I'm not a racist, honest!

Phyzzle posted:

Yes.

What are the odds that the last coin flip is heads? What are the odds that one of the hundred coin flips is heads?

That second question isn't quite the same though. Asking: "if I flip a coin 100 times, what is the probability that any is heads?" is not the same as "if I flip a coin 100 times and 99, in any order, are tails, what is the probability that the other one is heads?" In the second instance, for 99 of the flips, the probability of heads is now 0, which changes the overall total a bit.

JVNO posted:

No, he's absolutely right about 67%. Trust me when I say statistics are often incredibly unintuitive- but you really need to take a course on Bayesian inference to understand the key difference between Bob and Bill. Prior information is key- which is why this is superficially similar to Monty Hall.

I'll admit to being ignorant of the nuances of Bayes' Theorem, but I'm not just intuiting my answer here, I'm at least attempting to run the numbers myself. The way it's being phrased is giving us a false impression of what our possibilities are. Obviously the probability of one child being a boy is (roughly) 50/50. Adding the second child gives us the four combinations. That much I can grant. But in both cases, we know we can eliminate one child from consideration, because we're told the gender. We don't have to consider that one anymore when assessing the probability of the other. That's why I asked the 100 children question; the probability of the other 99 being boys is 0, because we are told they are girls. It's functionally equivalent to "I have 100 children, but let's ignore 99 for a moment - consider just this one child, what is the probability they are a boy?"

Another way to think about it: we have the four combinations - BB, BG, GB, GG - each with a 25% probability, and we have information we can use to eliminate some of our choices. In the first, this is easy, because we are told something about birth order, so we take away BB and GB, leaving us with the other two. In the second, we aren't told order, so it feels like GB and BG are both still on the table, giving us 3 equally likely outcomes. This isn't the case, though; one of those two options is impossible, because there is definitely a girl, we're just not sure where. If the girl is the older sibling, the possibility that the older sibling is a boy is 0, even if we don't know that. True, we can't know which choice between BG and GB is impossible, but knowing that one of them is impossible is enough. We can know that BB and [one of (BG or GB)] are eliminated, giving us 2 options.

It kinda reminds me of Sudoku, where you know the 4 is in one of like three boxes, and you don't know which, but you can still use that knowledge to eliminate the possibility of a 4 being in some other boxes that are still blank.

To again draw the comparison to Monty Hall, this is like saying there's two doors, and both can have either a goat or a car. So there might be two cars, might be two goats, might be one of each. Monty then opens a door to reveal what's behind it, and asks you the odds of the other door being a car. If we specified that he opened Door #1, rather than "either door", that would not affect the 50/50 chance of the car being behind the door that's been left unopened.

The feeling that there's 3 equally likely outcomes in the latter case seems more of a trick of language rather than the quirks of statistical mathematics. Still, this isn't a hill I'm willing to die on; I can certainly be talked out of this.

GAINING WEIGHT...
Mar 26, 2007

See? Science proves the JewsMuslims are inferior and must be purged! I'm not a racist, honest!

A Buttery Pastry posted:

That's not how it works. You can't eliminate an option because it's incompatible with another option, because they're all incompatible with each other. It can be either BG, GB, or GG, all of which are mutually exclusive, not just BG and GB, leaving you with only a single (undetermined) combination. Which leads us back to the 1/3 chance for any of them being the right one.

I think the main disagreement seems to be between whether our scenario is this:

[1] Three equal choices:
GG
-or-
GB
-or-
BG

or this:

[2] Two equal choices:
GG
-or-
[one of:(GB or BG)]

The reason I think it's 2 and not 1 is this: you're right that GB and BG are equally likely to each other, but they aren't equally likely to GG. It's like a nested 50/50 chance within another 50/50 chance. That's why I was going the "collapse" route. We don't know which is possible, but we know if one is possible (that the girl we're told about is the younger sibling and her older sibling is either a boy or a girl) then the other is impossible (that the girl we're told about it the older sibling).

We could take the opposite route, and instead of collapsing the one-of-each possibility, expand the both-girls possibility. We're told that one sibling is a girl, but not which one. It could be the older or the younger. This actually gives us two outcomes where both siblings are girls: the girl we were told about is the older one, who has a younger sister, -or- the girl we were told about is the younger sibling, and she has an older sister. Which we could show like this:

GG -or- GG

With the underlined G being the sibling we were told about. So there's actually four possibilities, given the knowledge that there are two children, one of whom is a girl:

GG
GG
GB
BG

Thus the probability of us having a boy somewhere in the mix is 2/4, or 50%. That's why I'm saying it's a trick of language: we have GG as one possibility because we see no difference in GG or GG because they are both represented by the word "girl". It's actually two possibilities hidden in one, masked by the word we use to refer to their gender.

quote:

Any combination with only one heads (Fertile Bill scenario):
HTT...T
THT...T
TTH...T and so on until
TTT...H

Plus a final "no heads combination"
TTT...T

Leaving you with 100 options where you have exactly one heads, and one option with no heads, resulting in a 100/101 chance of there having been 1 heads if 99 of a 100 are tails, or ever so slightly above 99% chance.

Phyzzle posted:

It is the same. For 99 of the flips, the probability of heads is now 0, which does not change the probability of the other flip.

"If I flip a coin 100 times, and 99, in any order, are tails, what is the probability that #1 is heads Or #2 is heads Or #3 is heads Or #4 is heads Or #5 is heads Or . . . "

To put a finer point on my above reasoning with the 100 coins example, your treating of "no heads" as one option is mistaken. It's actually 100 options: the coin we didn't know about is the first, and it was tails; the coin we didn't know about was the second, and it was tails; etc. There's 100 cases where the "other coin" was tails, and 100 cases where it was heads, giving us a 100/200 overall chance of heads and a 100/200 overall chance of tails.

GAINING WEIGHT...
Mar 26, 2007

See? Science proves the JewsMuslims are inferior and must be purged! I'm not a racist, honest!

Phyzzle posted:

See the 'Or' in the statement about one heads result? You're statement about no heads has semicolons, but they don't represent 'Or', do they? They represent 'And'. A bunch of 'And's strung together to make one unlikely case, instead of a bunch of cases connected by 'Or's to make one much more likely case.

Of course they mean or?

Buried alive posted:

You're being inconsistent. Yes, if BG is the case then GB is not the case. However, we don't even know if Bill has a boy at all. If GG is the case, then neither BG nor GB are the case. Similarly, if it turns out that BG is the case, then neither GB nor GG are the case. Two of the three options are impossible, depending on what the result is. Which results in this:
[one of:(GG or GB or BG)]. Which is just scenario 1, phrased differently.

I am fairly sure you're the one being inconsistent. We can talk about two possibilities ("contracted" view as above), which are: either both are girls, or there is one of each; or we can talk about four possibilities ("expanded" view), where order matters and GG is two of the possibilities, but we can't talk about 3 outcomes, because then you're expanding/contracting one and not the other.

This is easier to see if neither of the two options for the unknown child match the first. So let's add in a third gender and phrase the problem like this: there are two kids, one is a girl, the other is either a boy or a flibbertyjib. Then it's easy to see that there are four possibilities (taking birth order into account) :

GB
GF
BG
FG

Half of which are B, giving us the 50%

quote:

Which sibling we're told about doesn't matter. The thing we're being asked to consider is which set of circumstances is true? The 1st being a girl and the 2nd being a girl is the same set of circumstances regardless of which sibling we're told about. The 1st being a boy and the 2nd being a boy is a different set of circumstances due to which sibling it actually is, not which one we're told about.

Right, this is the inconsistency I was talking about. Either we're considering birth order or we're not, we can't do it for the second and not the first.

A Buttery Pastry posted:

Would you say the chance of getting two sixes when rolling two dice, is the same as getting a six and a one?

Yes. The second case seems more likely because you have 2 options per die, but you have to subtract out the cases where you get 2 sixes or 2 ones, which is not the outcome you're looking for. Both are 1/36.

quote:

No, there is only a single combination of flips that make 100 out of 100 coins become tails, this one:

TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT

This is the trick of language thing. There are 100 cases where a single coin is unknown, then it turns out to be tails. The fact that it matches the state of the 99 coins is confusing, and makes it seem like it's only 1 option. See the "third gender" explanation above and apply it thusly.

GAINING WEIGHT...
Mar 26, 2007

See? Science proves the JewsMuslims are inferior and must be purged! I'm not a racist, honest!

Phyzzle posted:

So the "it" in the sentence refers to coin we don't know about? Then you're assuming that probability of it being tails is 1/2, so that the sum is 100/200 for all coins. But assuming the 1/2 begs the question.

I'm sorry, I don't follow. Why would the probability of a coin being tails being 1/2 be begging the question? We're assuming fair coin right?

botany posted:

answer A: can we please play poker some time.

answer B: https://www.geogebra.org/m/UsoH4eNl

I'll go ahead and concede the dice thing, but it's not really the same question anyway. I'm really only still persisting on the boy/girl problem, and I think the "third gender" explanation is my best articulation of why I don't find the 1/3 answer convincing.

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GAINING WEIGHT...
Mar 26, 2007

See? Science proves the JewsMuslims are inferior and must be purged! I'm not a racist, honest!

VitalSigns posted:

Let's brute-force the boy-girl problem. I wrote a MATLAB script
code:
%initialize cases. boy is total outcomes of at least one boy. noboys is total outcomes of no boys (all girls)
boy=0;
noboys=0;
index=0;
while(index<10000)
%loop 10,000 times
     %set each child to a random integer from 0 to 1. 0 will be boys, 1 will be girls.
     c1=randint(1,1,[0 1]);
     c2 = randint(1,1,[0 1])

     if(c1+c2==2)
     %this means both children were girls, increment noboys. Increment loop index.
          noboys=noboys+1;
          index =index+1;
     else if(c1+c2==1)
     %this means one child is a boy, increment boy, increment loop index.
          boy=boy+1;
          index=index+1;
          end
     end
%the other case, c1+c2 == 0 means both children were boys, ignore this case because it's impossible as given by the problem. Do not increment loop index.
end
boy_percent=boy*100/10000;
noboys_percent=noboys*100/10000;


66.98% instances of a boy, 33.02% instances of only girls.

Fair enough, I can't argue with the brute numbers. I wonder what was at fault with my "third gender" conception - seems like it should come out the same, yeah? It just uses a different word for "girl that isn't the child the man was talking about", right?

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