Stopping rules and gender ratios

In a society, every couple wants a girl. Each child they have is equally likely to be a boy or a girl, independently. Couples keep having children until they get a girl, then stop.

Build a family

Each child is 50/50 boy or girl. The couple keeps going until they get a girl.

What do you think?

If every couple stops after having a girl, what fraction of the population will be girls?

Why stopping rules don't matter

The misconception is that since every family ends with a girl, girls must outnumber boys. Look at what families actually produce:

With probability 1/2: family is G (1 girl, 0 boys)
With probability 1/4: family is BG (1 girl, 1 boy)
With probability 1/8: family is BBG (1 girl, 2 boys)
With probability 1/16: family is BBBG (1 girl, 3 boys)
In general, with probability $1/2^k$ : family has $k-1$ boys and 1 girl

Stopping rule irrelevance

A stopping rule decides when to stop observing, but it cannot change the distribution of each observation. If each birth is independently 50/50, no rule about when to stop having children can shift the overall ratio away from 50/50.

The math

Each family type contributes "number of boys × probability of that family type." From the list above: G has 0 boys with probability 1/2, BG has 1 boy with probability 1/4, BBG has 2 boys with probability 1/8, and so on. Adding these up gives the expected number of boys per family:

$E[\text{boys}] = 0 \cdot \Large\frac{1}{2} + 1 \cdot \Large\frac{1}{4} + 2 \cdot \Large\frac{1}{8} + 3 \cdot \Large\frac{1}{16} + \cdots = \Large\sum_{k=0}^{\infty} k \cdot \Large\frac{1}{2^{k+1}} = 1$

The expected number of girls per family is exactly 1 (every family has exactly one girl).

So the expected ratio of girls to total children is $1/(1+1) = 1/2$ .

Computing E[boys]

E[\text{boys}] = \sum_{k=0}^{\infty} k \cdot \frac{1}{2^{k+1}}

A family with k boys has probability (1/2)^(k+1): k heads followed by a tail.

Step 1 of 3

Family simulator

Run a few thousand families. The girl percentage stays near 50% regardless of sample size.

The stopping rule controls family size, not the gender of any individual child. Each birth is an independent coin flip. No amount of strategic stopping can change the underlying 50/50 odds. The same logic applies to gamblers who try to "quit while ahead" — the strategy affects when you stop, not the expected outcome.

What do you think?

What if couples stop after having TWO girls? Does the girl fraction change?

Practice

Under the 'stop after first girl' rule, what's the expected family size (total children)? (whole number)

1/2