The Country That Wanted More Boys

The problem
keep having children until a boy · then stop?????????what is the long-run boy-to-girl ratio?

A country wants more boys. The government decrees:

Every family must keep having children until they have a son. Then they stop.

Assume each birth is independently a boy or girl with probability 1/21/2. Over a long time, what is the country's boy-to-girl ratio?

Tempting (but wrong)
two opposite intuitionsMore boys?every familyends in a boyMore girls?long girl-streakfamilies pile upboth feel right · both wrong

Two opposite intuitions are common:

  • "More boys." Every family ends with a boy, so boys must dominate.
  • "More girls." Some unlucky families have 4, 5, 6 daughters before a son finally arrives — those long girl streaks pile up.

Both arguments sound persuasive. Both are wrong.