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Buried alive
Jun 8, 2009

GAINING WEIGHT... posted:

...
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.
...

GAINING WEIGHT... posted:

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

or this:

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

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.

GAINING WEIGHT... posted:

...
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.

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.

Buried alive fucked around with this message at 16:01 on Sep 4, 2016

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Buried alive
Jun 8, 2009
Edit: I'm a dumb.

Buried alive
Jun 8, 2009

wateroverfire posted:

Hear me out, man, and follow the logic below to the end.

I know this looks intuitive but it's wrong. TH and HT are outcomes that become mutually exclusive when information is added about the outcome of one of the coins - even if we don't know which coin we have information about.

So say you tell me "at least 1 coin is T" and show me a T, so we end up with the table of results in your quote above.

1) If Coin 1 is T, the only possible outcomes are TH and TT. This is correct and should also be intuitive if you think about it, but if you disagree then show how and let's talk it through.

2) If Coin 2 is T, the only possible outcomes are HT and TT by the same reasoning.

Therefore, the probability that we are in a world in which the other coin is H is the following:

P(Coin you showed is Coin 1) * P(TH conditional on Coin1 being T) + P(Coin you showed is Coin 2)*P(HT conditional on Coin 2 being T)

We have only two coins so

P(Coin you showed is Coin 1) = 50%
P(Coin you showed is Coin 2) = 50%

Following 1), P(TH conditional on Coin1 being T) = 50%
Following 2), P(HT conditional on Coin 2 being T) = 50%

Plug everything in and 50%*50% +50%*50% = 25% + 25% = 50% probability the other coin is an H. Proving what we know from having defined the coin flips as independent events.

The above is NOT the Monty Hall problem in concept.

In Monty Hall, the values of the "flips" are contingent - there is only 1 Car and the other two "flips" must be Goat. So revealing a Goat gives you information about the potential values of the other "flips" where revealing a T in the coin flip problem gives you nothing.

edit: Notice, if you work through the formulas, it doesn't matter how I come by the information that one coin is T.

If you pick a coin at random and it happens to be T, the results tables are the same and the probability calculation is the same 50%.

If you deliberately show me a T, the results tables are the same and the probability calculation is the same 50%.

If you deliberately show me Coin 1 that happens to be a T, the results tables are the same and the P(Coin you showed me is Coin 2) is 0% while the P(Coin you showed me is Coin 1) is 100%. The probability calculation works out to the same 50%.

If you deliberately show me Coin 2 that happens to be a T, the results tables are the same and the P(Coin you showed me is Coin 1) is 0% while the P(Coin you showed me is Coin 2) is 100%. The probability calculation works out to the same 50%.

This is literally the exact same reasoning as this:

GAINING WEIGHT... posted:

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.



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.

He was wrong then, you are wrong now:

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.

GAINING WEIGHT... you should probably listen to the people telling you why your intuition is wrong.

Edit: alternatively play the coinflip game with me a bunch of times.
Every time you flip both heads, I'll give you a 2.2:1 payout instead of the 2:1 payout that would make it an even game according to you. You can't lose!

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