(ii) Let Y be the random variable

*number of boys in a family*.

Y = 2:

BB. Pr(Y = 2 | X = 2) = 1/4.

BGB or GBB. Pr(Y = 2 | X = 3) = 1/8 + 1/8 = 1/4.

GBGB, GGBB or BGGB. Pr(Y = 2 | X = 4) =

**3/16**.

Therefore Pr(Y = 2) = 11/16.

Y = 1:
BGGG, GBGG, GGBG or GGGB. Pr(Y = 1 | X = 4) =

** 1/4**.

Therefore Pr(Y = 1) = 1/4.

Y = 0:
GGGG. Pr(Y = 0 | X = 4) = 1/16.

Therefore Pr(Y = 0) = 1/16.

E(Y) = (0)(1/16) + (1)(1/4) + (2)(11/16) = 26/16 = 13/8.

Since E(X) = 13/4, the expected number of girls is 13/4 - 13/8 = 13/8.

This makes sense since, if you let Z be the random variable

*number of girls in a family*:

Pr(Z = 0) = Pr(X = 2) = 1/4.

Pr(Z = 1) = Pr(X = 3) = 1/4.

Pr(Z = 2) = 3/16. (GBGB, GGBB or BGGB).

Pr(Z = 3) = Pr(Y = 1) = 1/4.

Pr(Z = 4) = 1/16. (GGGG)

E(Z) = (0)(1/4) + (1)(1/4) + (2)(3/16) + (3)(1/4) + (4)(1/16) = 13/8.

So I get the ratio to be 13/8 : 13/8 <=> 1:1.