My statistical response to liuigi’s post

I was going to make this a comment, but I spent enough time thinking about it that I figured it deserved it’s own post.
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Q2 depends on how you interpret “you find out that one of their children is male”. Suppose that you discovered this information when you asked a parent “Do you have any boys?” To which she responds “Yes, we have at least one boy”. Despite the fact that you’d probably be slightly scared by the precise way in which the mother responded, you now know that “at least one of the children is male”, which you could also express as “one of their children is male”
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Statistically, suppose you have a random variable X, where X means “The number of male children in a family of two children”. X is a discrete random variable with a binomial distribution.
P(X = 0) = 0.25
P(X = 1) = 0.50
P(X = 2) = 0.25
If you interpret “you find out that one of their children is male”, to mean “At least one child is male”, we can specify this condition as X > 0.
So by bayes theorem, P(X = 2 | X > 0) = P(X = 2) / P(X > 0) = .25/.75 = 1/3
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However, there is another way to interpret “you find out that one of the children is male”. You could interpret this as “You met a specific child, and he was male”. In this scenario, there are two children behind a closed door. One of them comes out, and he is male. What is the probability that the other child, still behind the closed door, is male?
Now, we can express each child as a bernouli random variable. X1 is a bernoili random variable that means “the first child you meet is male” and X2 is a bernouli random variable that means “the second child you meet is male”. So we have P(X1) = 0.5, and P(X2) = 0.5. X1 and X2 are assumed to be independant. Therefore, we have P(X2 | X1) = 0.5, so the probability that the second child is male is 50%.
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While you probably agree with me at this point that the statement “you find out that one of the children is male” can be interpreted two different ways, you may not agree with me that those two statements actually convey different amounts of information. “I have at least one male son” seems to mean the same thing as “This specific son is male”. Lets change the format of the problem. Suppose two coins have been flipped, and placed underneath cups. If I point to a cup, and say “Under this cup, a coin landed on heads”, does this give you any information whatsoever about the coin under the other cup? Of course not! Yet if for those same two cups, I tell you “At least one of these cups has a coin with a head underneath it”, the situation is completely different. In this scenario, you can most definately use the method that Chris described, since this would be a problem where you would use a binomial random variable to determine the probability.
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Edit: Since Liu didn’t think I was being argumentative enough, I’ll say for the record that I think it would be extremely odd if a parent said “Yes, I have at least one son”

4 comments

  1. [15:05] Chris: so now my cute kittens have been displaced by your bernoulli variables
    [15:05] Chris: which makes me sad
    [15:05] Tim__ Gas: bwhahaha

    This is what we get for demanding mathematical precision. Obfuscation of information, which leads to intellectual aristocracy. Tyranny, I tell you.

  2. It’s simply an example of the inprecise nature of spoken language. I’m just going to start being more precise with my english. Now, whenever I use an “or”, I’ll be specific as to whether I mean inclusive or exclusive or. “Would you like Soda XOR Water to drink?”

  3. Here we go.

    Lets just say my name is Wilt Jr.(its actually not), probably bonus points if flakes or gas know why. I have one male. I receive a letter from a past girlfriend saying I have a kid. Lets just say its not some random person trying to get into my cash stash. This would fit our problem and it would not be strange to say, very honestly, “Yes, I have at least one son.”. I won’t mention the tranny on the spot idea, even though i just did.

    Just to clear up any confusion, Chengameepheus made up that example.

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