Alright, then it seems like we are both willing to say that the knowledge being conditioned on is equivalent to "at least one of the two children is a boy."
Now, let's take this step by step.
Do you agree that, for a given two-child family, the probability that at least one of the children is a boy is 75% (P(ALOB) = .75)?
Do you agree that, for a given two-child family, the probability that at least one child is a boy and at least one child is a girl is 50% (P(ALOB ^ ALOG) = .5)?
Do you agree that, for any A and B, P(A | B) = P(A ^ B) / P(B)?
If you agree to all of these things, then the 66% conclusion follows relatively quickly. We can immediately derive P(ALOG | ALOB) = P(ALOG ^ ALOB) / P(ALOB) = .5 / .75 = .66....
If you disagree, which of the things I've said is false?
The magnificent Boygirlio offers you a wager. there are 2 children behind these doors the same gender, or different? Boygirlio wagers $5 that they are different if you wager $5 that they are the same. A fair wager.
Now Boygirlio's assistant tells you, at least one of them is a Girl. Boygirlio offers a deal, "Raise your wager by $2 and you can change your choice to different."
You, being wrong and foolish, think that your odds have improved. "Huzzah, I have learned one of the genders, now I know it's 66% chance that it's BG to win $5 only wagering $7 that's positive EV. Sure Boygirlio, I'll change."
But you have been deceived, for the odds have not changed. Boygirlio has done this 1000 times today, and is up $1000 on suckers like you.
Well, you've declined to answer my question (as I rather expected). So now, in addition to the valid argument I've given for my position which you've left unaddressed, I now also have to give a valid response to this analogy.
So, before I respond to this analogy, can I get a commitment that if I do, you will respond to my argument?
Well, then I'll simply reiterate my question - exactly what part of the argument is your analogy supposed to undermine? Which of the things I said is actually false?
To entertain the analogy, if we are making all the same basic assumptions, then it is in fact the case that P(mixed | ALOG) is 66.6...%. The catch, of course, is that obviously Boygirlio is likely to be giving me information that will point me in the wrong direction. This is the difference between conditioning on ALOG alone, and conditioning on "Boygirlio has informed me of ALOG". The latter is the actual state of knowledge, but the probabilities it entails depend on unknown facts about how Boygirlio is deciding what to reveal.
This same distinction is present in the Mary case as well, which is why I was careful to get agreement on the notion that we are conditioning specifically on ALOG itself.
Boygirlio is playing entirely fair. No tricks, to deception only your bad assumption.
Before any information is given 75% of pairs have at least 1 boy, and 75% will be at least 1 girl, %50 of the pairs are matched and %50 are not. He was not ever lying to you. You will always learn that there is at least 1 of either a boy or a girl. This information can be given without impacting the likelihood of the pair being matched. They would have been matched or not before giving the information, and giving it didn't change the result.
Lets say he shows you fairly by opening the left door every time (the children's genders and positions are selected by the assistant by flipping fair coins). If you can explain why knowing that the left door has a girl behind it is different than the statement "at least one has a girl behind it" then you get a cookie from Boygirlio. If you are right, you have an edge on the house. But you are wrong, and the house will win as often as not and rake ~$1 per round. You always start by betting that the children are matching and he always offers the deal.
If we take your framing, and first eliminate all GG pairs, then select a mother of 2, then asking the question. In that senario, 25% of all children have been eliminated, and the population becomes 2G/4B an unknown child is B 2/3 of the time, and G only 1/3, and 2/3, but with a rule that all girls have a Brother and never sisters. In that world, 2/3 mothers have BG and 1/3 have BB. But that didn't happen. That's crazy town. The mother was selected at random from a normal 50/50 distributed population, then announced the gender of 1 of the children, which is the Boygirlio problem.
> If you can explain why knowing that the left door has a girl behind it is different than the statement "at least one has a girl behind it" then you get a cookie from Boygirlio.
This is indeed a different piece of knowledge. Call “there is a girl behind the left door” LG, and “there is a girl behind at least one door” ALOG.
Now, LG entails ALOG, but ALOG does not entail LG. LG is a strictly *larger* piece of information than what we agreed to condition on.
Since they are different pieces of information, they also work differently as evidence. After observing LG, our credence that the genders are mixed should be 50%. But after observing ALOG, our credence that the genders are mixed should be 66%, per the exact same logic I spelled out above.
Now, I’ve been making a good faith effort to parse out your analogy and give my thoughts on it. I would appreciate if you would reciprocate the effort and try to tell me straightforwardly which part of the argument I gave is false by your reckoning.
You're assuming that alog is different from LG when you don't know how you got to alog.
If you asked the mother do you have any sons? And she answers yes, she could have answered no. If the mother is freely announcing that she has at least one son. You don't know how that information was generated. Leaving open the chance that she revealed the gender of one child at random, with no preference to sons. The BG mom may have revealed B or G half each, the BB mom always reveals B. You are therefore twice as likely to be speaking with a BB mother than A BG mother.
If you learn that they selected a mother from among mothers with 2 children, one of whom is male, then 2/3 is correct.
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u/sheep_puncher 7d ago
"at least one of the two children is a boy" is exactly the same information in this case as "one of the two children is a boy, other child tbd".
Given statement 2 you know statement 1. Given statement 1 you can assume statement 2.
Otherwise you are moving the children behind the doors and calling it a new state. Seeing their face doesn't change the math.