I have the following questions regarding probability theory:
In the country Malthusia, every married couple has precisely four children.
1) Assume sons and daughters are born with equal probability and independent of the other children. Compute the expectation and variance of the number of girls in a family.
If we take $B=boy$ and $G=girl$ then we have $5$ possible outcomes (I think) if the sequence does not matter, i.e., we have $BBBB$, $BBBG$, $BBGG$, $BGGG$, $GGGG$. Each with probability $P$ $=$ $\frac{1}{2}^4$ $=$ $\frac{1}{16}$, because of the independancy. I was thinking of the following $E[G] = np = 4 * 0.5 = 2$ and $Var(G) = np(1-p) = 4*0.5*0.5 = 1$
2) Assume sons and daughters are born with equal probability and independent of the other children. A child is selected at random from a family and is found to be a boy. Compute the expected number of his sisters and brothers.
$BBBB$ $\rightarrow$ $P$ $=$ $\frac{1}{16}$. Pick a boy gives probability $1$.
$BBBG$ $\rightarrow$ $P$ $=$ $\frac{1}{16}$. Pick a boy gives probability $0.75$.
$BBGG$ $\rightarrow$ $P$ $=$ $\frac{1}{16}$. Pick a boy gives probability $0.50$.
$BGGG$ $\rightarrow$ $P$ $=$ $\frac{1}{16}$. Pick a boy gives probability $0.25$.
$GGGG$ $\rightarrow$ $P$ $=$ $\frac{1}{16}$. Pick a boy gives probability $0$.
3) Assume that the first three children are born with equal probability and independent of the other children. However, the fourth child has the same sex as the third child with probability 1/4. A child is selected at random from a family and is found to be a boy. Compute the expected number of his sisters and brothers.
Any help with one of these 3 questions would be beautiful. Thanks in advance.