Solution: 2019 Winter Final - 22

Author: Michiel Smid

Question

When a couple has a child, this child is a boy with probability 1/2 and a girl with probability 1/2, independent of the gender of the other children. A couple stops having children as soon as they have 2 girls or 2 boys. Consider the random variables

G = the number of girls the couple has,

and

B = the number of boys the couple has.

Which of the following is correct?

(a)

None of the above.

(b)

The random variables $G$ and $B$ are independent.

(c)

The random variables $G$ and $B$ are not independent.

(d)

All of the above.

Solution

Let’s just check if it’s true for some $ \PR(G=x \cap B=y) = \PR(G=x) \cdot \PR(B=y) $

Determine $\PR(G=0)$

They have 0 girls

This happens when 2 boys pop out one after the other

$\PR(G=0) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$
Determine $\PR(B=0)$

They have 0 boys

This happens when 2 girls pop out one after the other

$\PR(B=0) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$
Determine $\PR(G=0 \cap B=0)$

They have 0 girls and 0 boys? This never happens. They need to get preggers and have babies

$\PR(G=0 \cap B=0) = 0$
Profit

$\PR(G=0 \cap B=0) = \PR(G=0) \cdot \PR(B=0)$

$0 \neq \frac{1}{4} \cdot \frac{1}{4}$

Therefore, the random variables $G$ and $B$ are not independent.