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EHeller comments on Bayes' Theorem Illustrated (My Way) - Less Wrong
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1/3 (you either got hh, heads/tails,or tails/heads). You didn't tell me THE FIRST came up heads. Thats where you are going wrong. At least one is heads is different information then a specific coin is heads.
This is a pretty well known stats problem, a variant of Gardern's boy/girl paradox. You'll probably find it an intro book, and Jiro is correct. You are still overcounting. Boy-boy is a different case then boy-girl (well, depending on what the data collection process is).
If you have two boys (probability 1/4), then the probability at least one is born on Tuesday (1-(6/7)^2). ( 6/7^2 being the probability neither is born on Tuesday). The probability of a boy-girl family is (2*1/4) then (1/7) (the 1/7 for the boy hitting on Tuesday).
Lets add a time delay to hopefully finally illustrate the point that one coin toss does not inform the other coin toss.
I have two coins. I flip the first one, and it comes up heads. Now I flip the second coin. What are the odds it will come up heads?
No one is suggesting one flip informs the other, rather that when you say "one coin came up heads" you are giving some information about both coins.
This is 1/2, because there are two scenarios, hh, ht. But its different information then the other question.
If you say "one coin is heads," you have hh,ht,th, because it could be that the first flip was tails/the second heads (a possibility you have excluded in the above).
No, it's the exact same question, only the labels are different.
The probability that any one child is boy is 50%. We have been told that one child is a boy, which only leaves two options - HH and HT. If TH were still available, then so would TT be available because the next flip could be revealed to be tails.
Here's the probability in bayesian:
P(BoyBoy) = 0.25 P(Boy) = 0.5 P(Boy|BoyBoy) = 1
P(BoyBoy|Boy) = P(Boy|BoyBoy)*P(BoyBoy)/P(Boy)
P(BoyBoy|Boy)= (1*0.25) / 0.5 = 0.25 / 0.5 = 0.5
P(BoyBoy|Boy) = 0.5
It's exactly the same as the coin flip, because the probability is 50% - the same as a coin flip. This isn't the monty hall problem. Knowing half the problem (that there's at least one boy) doesn't change the probability of the other boy, it just changes what our possibilities are.
No, it isn't. You should consider that you are disagreeing with a pretty standard stats question, so odds are high you are wrong. With that in mind, you should reread what people are telling you here.
Now, consider "I flip two coins" the possible outcomes are hh,ht,th,tt
I hope we can agree on that much.
Now, I give you more information and I say "one of the coins is heads," so we Bayesian update by crossing out any scenario where one coin isn't heads. There is only 1 (tt)
hh,ht,th
So it should be pretty clear the probability I flipped two heads is 1/3.
Now, your scenario, flipped two coins (hh,ht,th,tt), and I give you the information "the first coin is heads," so we cross out everything where the first coin is tails, leaving (hh,ht). Now the probability you flipped two heads is 1/2.
I don't know how to make this any more simple.
http://en.wikipedia.org/wiki/Boy_or_Girl_paradox
I know it's not the be all end all, but it's generally reliable on these types of questions, and it gives P = 1/2, so I'm not the one disagreeing with the standard result here.
Do the math yourself, it's pretty clear.
Edit: Reading closer, I should say that both answers are right, and the probability can be either 1/2 or 1/3 depending on your assumptions. However, the problem as stated falls best to me in the 1/2 set of assumptions. You are told one child is a boy and given no other information, so the only probability left for the second child is a 50% chance for boy.
Did you actually read it? It does not agree with you. Look under the heading "second question."
I did the math in the post above, enumerating the possibilities for you to try to help you find your mistake.
Edit, in response to the edit:
Which is exactly analogous to what Jiro was saying about the Tuesday question. So we all agree now? Tuesday can raise your probability slightly above 50%, as was said all along.
And you are immediately making the exact same mistake again. You are told ONE child is a boy, you are NOT told the FIRST child is a boy. You do understand that these are different?
Re-read it.
The relevant quote from the Wiki:
We have no general population information here. We have one man with at least one boy.
I'm not at all sure you understand that quote. Lets stick with the coin flips:
Do you understand why these two questions are different: I tell you- "I flipped two coins, at least one of them came out heads, what is the probability that I flipped two heads?" A:1/3 AND "I flipped two coins, you choose one at random and look at it, its heads.What is the probability I flipped two heads" A: 1/2
For the record, I'm sure this is frustrating as all getout for you, but this whole argument has really clarified things for me, even though I still think I'm right about which question we are answering.
Many of my arguments in previous posts are wrong (or at least incomplete and a bit naive), and it didn't click until the last post or two.
Like I said, I still think I'm right, but not because my prior analysis was any good. The 1/3 case was a major hole in my reasoning. I'm happily waiting to see if you're going to destroy my latest analysis, but I think it is pretty solid.
Yes, and we are dealing with the second question here.
Is that not what I said before?
We don't have 1000 families with two children, from which we've selected all families that have at least one boy (which gives 1/3 probability). We have one family with two children. Then we are told one of the children is a boy, and given zero other information. The probability that the second is a boy is 1/2, so the probability that both are boys is 1/2.
The possible options for the "Boy born on Tuesday" are not Boy/Girl, Girl/Boy, Boy/Boy. That would be the case in the selection of 1000 families above.
The possible options are Boy (Tu) / Girl, Girl / Boy (Tu), Boy (Tu) / Boy, Boy / Boy (Tu).
There are two Boy/Boy combinations, not one. You don't have enough information to throw one of them out.
This is NOT a case of sampling.
"The first coin comes up heads" (in this version) is not the same thing as "one of the coins comes up heads" (as in the original version). This version is 50%, the other is not.
How is it different? In both cases I have two independent coin flips that have absolutely no relation to each other. How does knowing which of the two came up heads make any difference at all for the probability of the other coin?
If it was the first coin that came up heads, TT and TH are off the table and only HH and HT are possible. If the second coin came up heads then HT and TT would be off the table and only TH and HH are possible.
The total probability mass of some combination of T and H (either HT or TH) starts at 50% for both flips combined. Once you know one of them is heads, that probability mass for the whole problem is cut in half, because one of your flips is now 100% heads and 0% tails. It doesn't matter that you don't know which is which, one flip doesn't have any influence on the probability of the other. Since you already have one heads at 100%, the entire probability of the remainder of the problem rests on the second coin, which is a 50/50 split between heads and tails. If heads, HH is true. If tails, HT is true (or TH, but you don't get both of them!).
Tell me how knowing one of the coins is heads changes the probability of the second flip from 50% to 33%. It's a fair coin, it stays 50%.
Flip two coins 1000 times, then count how many of those trials have at least one head (~750). Count how many of those trials have two heads (~250).
Flip two coins 1000 times, then count how many of those trials have the first flip be a head (~500). Count how many of those trials have two heads (~250).
By the way, these sorts of puzzles should really be expressed as a question-and-answer dialogue. Simply volunteering information leaves it ambiguous as to what you've actually learned ("would this person have equally likely said 'one of my children is a girl' if they had both a boy and girl?").
Yeah, probably the biggest thing I don't like about this particular question is that the answer depends entirely upon unstated assumptions, but at the same time it clearly illustrates how important it is to be specific.