I've got the following question:
We have a family with 3 children and every child has the same probability for male or female. What is the chance for the youngest child being female, if we know
a) the oldest one is female
b) we have a least one female child
c) we have exactly one female
I tried to solve it we conditional probability:
a) $P(X_1=M|X_3=M)=\frac{P(X_1=M,X_3=M)}{P(X_3=M)}=\frac{\frac{1}{2} \cdot \frac{1}{2}}{\frac{1}{2}}=\frac{1}{2}$
b) $P(X_1=M \vee X_2=M \vee X_3=M|X_3=M)=\frac{X_1=M \vee X_2=M \vee X_3=M}{X_1=M}=\frac{\frac{7}{8} \cdot \frac{1}{2}}{\frac{1}{2}}=\frac{7}{8}$
c) $P(X_i=M|X_1=M)=\frac{\frac{3}{8} \cdot \frac{1}{2}}{\frac{1}{2}}=\frac{3}{8}$
Are my ideas correct and also my solutions?
Thanks a lot in advance.