Family Structure Oddity

New mom Alice decides she will have children until she has a boy immediately followed by a girl. Barb decides she wants a boy immediately followed by a boy.
Counterintuitively, and assuming the chances of having a boy or girl are the same, Alice will on average need to have four children to achieve her wish. On average Barb will need to have six. That's even though boy and girl births have an equal chance of occurring.

That boggles my mind, it's so counterintuitive.

Beware. If you're a math-phobe you might want to stop now. If brave enough to continue here's the basis.

The article Mathematicians Discover Prime Conspiracy refers in the section Prime Preferences to the equivalent counterintuitive finding by mathematician Tadashi Tokieda regarding coin tossing. It's explored in greater detail here. There's a discussion on Reddit.

I verified the result for myself with a random number generator and spreadsheet program.You can too. What you find is the distribution of the number of children to get to the preferred end point (boy-girl or boy-boy) is skewed. The median (half more, half less) is four in both cases. It's the average (mean) that's different.

I did warn you!

Is it true in real life? An examination of the structure of large families would be the test.

However, the analogy with coin tossing isn't exact as according to the CIA World Factbook as of 2014 the sex ratio at birth is estimated at 106 boys to 100 girls in Canada and 105 to 100 in the UK.

Going back to the spreadsheet program and incorporating the sex bias, with more boys born than girls Barb has a slightly improved chance of birthing two successive boys.

Who knew!