Here’s a puzzle based on something interesting that I learned from Greg Egan. I’ve dramatized it a bit.
Traditional Tom and Liberal Lisa are dating. They discuss their plans for having children:
Tom: I plan to keep having kids until I get two sons in a row.
Lisa: What?! That’s absurd. Why?
Tom: I want two to run my store when I get old.
Lisa: Even ignoring your insulting assumption that only boys can manage your shop, why in the world do you need two in a row?
Tom: From my own childhood, I’ve learned there’s a special bond between sons who are next to each other in age. They play together, they grow up together… they can run my shop together.
Lisa: Hmm. Well, then maybe I should have children until I have a girl followed directly by a boy!
Lisa: Well, I’ve observed that something special happens when a boy has an older sister, with no intervening siblings. They play together, they grow up together… and maybe he learns not to be such a sexist pig!
They decide they are incompatible, so they split up and each one separately tries to find a mate who will go along with their reproductive plan.
Now for some puzzles. In these puzzles, assume that each time someone has a child, they have a 50% chance of having either a daughter or a son. Also assume each event is independent: that is, the gender of any children so far has no effect on that of later ones. Also ignore twins and other tricky issues.
Puzzle 1. If Tom carries out his plan of having children until he has two consecutive sons, and then stops, what is the expected number of children he will have?
Puzzle 2. If Lisa carries out her plan of having children until she has a daughter followed directly by a son, and then stops, what is the expected number of children she will have?
Puzzle 3: Which is greater, Tom’s expected number of children or Lisa’s? Or are they equal?
For maximum benefit, try to answer Puzzle 3 before doing the calculations required to answer Puzzles 1 or 2.