[jspiral]

what works for me is to think of it like the mob of people is already there in front of you, and you're first grabbing out just a subset of the mob based on some conditions, then checking only those qualified at random which is the same as checking the distributions inside the qualified group. so for problem 1, the mob is all parents with 2 children, each child is either b or g. the underlying distribution is 25%bb, 25%gg, 50%bg.

you first cull the set by saying "only parents with at least 1 b, line up to be examined". i think it's clear that the selected population will be 1/3 bb parents, 2/3 bg parents, right?

on the other hand the second problem is you saying: "all parents with a b born on tuesday, report to be examined!". note that most of the parents that were in the question 1 selected population are now excluded. try to imagine which parents get selected by this one.

you can't make the inference back to the original question because it's a different question about a broader group of people. question 1 population distrubution actually is 1/3 vs 2/3. the key is to think of it as selecting different subsets of the parent mob.

[jldugger]

> on the other hand the second problem is you saying: "all parents with a b born on tuesday, report to be examined!". note that most of the parents that were in the question 1 selected population are now excluded. try to imagine which parents get selected by this one.

But we expect about the same number of parents to show up for "all parents with a b born on Tuesday" as "all parents with a b born on Monday." This feels like some kind of Simpson's paradox I've invented where every subgroup has a the same outcome, which is different than the outcome of the group as whole.

[deepserket]

If you have 2B you might show up in both groups (Monday and Tuesday in your example), if you have only 1B you can't show up in both groups, that's why the probability is different.

[jspiral]

true but note that the day of week selector is knocking out 6/7ths, it's quite a dominant excluder of parents. thus countering the effect of "has one boy already" on the composition of the group

[jldugger]

This is twisting my brain a bit -- there's a third possible version, where the only info you have is the day of week. Something like "i have a child, one of whom was born on tuesday. what are the odds the other is a boy?" perhaps.

And the answer there has to be basically 50:50. So when you combine these two filters, day of week wins somehow. Will have to ponder more

[jspiral]

ah i didn't actually explain it, the other responder did though. 6/7th (~85.7%) of parents with b g are knocked out by the day of week filter, while only ~%73.5 of the b b parents are knocked out by it. (if i did that math right) so proportionally the sample has more bb parents than the one without day of week.

[hgsgm]

And what makes probability so hard, is that these word problems describe a situation that never happens in real life, so human-language speaking people interpret the problem as something more sensible, before doing the math