On the other hand, if a couple told you they had five kids, at least four are boys, you would say to yourself, "it's not very likely that any couple would have five boys", and you would be correct...

You would be, as long as you don't say "it's therefore unlikely the fifth child is a boy".

Yes, BBBBB is rare. But BBBBG is just as rare (at least statistically. In practical terms, I suspect that having four boys in a row would be too exhausting and irritating, so they'd have given up by then).

Yeah, unless they have a lot of yard work they need done. Anyway, {BBBBG,BBBGB, BBGBB, BGBBB, GBBBB} is less rare.

> Yes, BBBBB is rare. But BBBBG is just as rare

Actually, it isn't. More boys are born than girls. (However, boys are somewhat more likely to die young.) Also, the odds for a given mating pair are not the same as the odds for the population as a whole.

And then there's post-conception sex-selection....

If the problem were "Given an ordered number of events (birth) what is the probability distribution of the last event". Then yes, it would be 50/50.

But you're looking for at least one boy. If you had 1000 children, intuitively, the probability of at least one being a boy is huge.