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<blockquote data-quote="2WS-Steve" data-source="post: 352721" data-attributes="member: 3289">The Ugly AnswerOkay, the formal solution to this gets kind of ugly since it relies on Bayes’ Theorem but if you’re interested here it goes:Bayes’ Theorem tells us the proper way to update our degrees of belief when presented with new evidence. Everything in this theorem is derivable from the standard probability axioms.Pr(H/E &amp; c) = [Pr(H/c)*Pr(E/H &amp; c)]/Pr(E/c)Pr(x) stands for the probability of xPr(x/y) stands for the probability of x given yH represents the hypothesis (i.e. that both children are the same sex)E represents the evidence (i.e. the photo of the male child)c represents the prior conditions (i.e. that probability space prior to the evidence); the prior conditions are just the state space described previously, i.e. the following options:1) M, F 2) M, M3) F, F4) F, M.We’re solving for Pr(H/E &amp; c) in this case…Pr(H/c) = .5; this is just the prior probability we assign before seeing the photograph.Pr(E/H &amp; c) = .5; this is the probability that we’ll actually see a photo of a male child given that the hypothesis is true and the background conditions. If the hypothesis is true we eliminate options 1 and 4 leaving us with just 2 and 3; 50% of those options shows a photo of a male.Pr(E/c) = .5; this is the probability of the evidence given the prior conditions. In other words, the straight probability of seeing a photo of a male child. Since there are 8 possible “photos” we can see (2 for each option) and 4 of those possibilities are male we end up with .5 here as well.That gives us:Pr(H/E &amp; c) = (.5 * .5)/.5 = .5---------------------Cool puzzle! The intuitive answer is the right one but figuring out why it’s right is the real puzzle.</blockquote>

[QUOTE="2WS-Steve, post: 352721, member: 3289"] [b]The Ugly Answer[/b] Okay, the formal solution to this gets kind of ugly since it relies on Bayes’ Theorem but if you’re interested here it goes: Bayes’ Theorem tells us the proper way to update our degrees of belief when presented with new evidence. Everything in this theorem is derivable from the standard probability axioms. Pr(H/E & c) = [Pr(H/c)*Pr(E/H & c)]/Pr(E/c) Pr(x) stands for the probability of x Pr(x/y) stands for the probability of x given y H represents the hypothesis (i.e. that both children are the same sex) E represents the evidence (i.e. the photo of the male child) c represents the prior conditions (i.e. that probability space prior to the evidence); the prior conditions are just the state space described previously, i.e. the following options: 1) M, F 2) M, M 3) F, F 4) F, M. We’re solving for Pr(H/E & c) in this case… Pr(H/c) = .5; this is just the prior probability we assign before seeing the photograph. Pr(E/H & c) = .5; this is the probability that we’ll actually see a photo of a male child given that the hypothesis is true and the background conditions. If the hypothesis is true we eliminate options 1 and 4 leaving us with just 2 and 3; 50% of those options shows a photo of a male. Pr(E/c) = .5; this is the probability of the evidence given the prior conditions. In other words, the straight probability of seeing a photo of a male child. Since there are 8 possible “photos” we can see (2 for each option) and 4 of those possibilities are male we end up with .5 here as well. That gives us: Pr(H/E & c) = (.5 * .5)/.5 = .5 --------------------- Cool puzzle! The intuitive answer is the right one but figuring out why it’s right is the real puzzle. [/QUOTE]

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