Question

A family has three children. Event $A$ is that family has at most one boy, Event ${}^{\prime }{B}^{\prime }$ is that family has at least one boy and one girl, Event ${}^{\prime }{C}^{\prime }$ is that the family has at most one girl. Find whether events ${}^{\prime }{A}^{\prime }$ and ${}^{\prime }{B}^{\prime }$ are independent. Also find whether $A,B,C$ are independent or not.

Answer 1

A family of three children can have
(i) All $3$ boys
(ii) $2$ boys+$1$ girl
(iii) $1$ boy+$2$ girls
(iv) $3$ girls
(i) $P$(3 boys)${=}^3{\text{C}}_0{\left(\frac{1}{2}\right)}^3=\frac{1}{8}$ (Since each child is equally likely to be aboy or a girl)
(ii) $P$ (2 boys+1 girl)${=}^3{\text{C}}_1\times {\left(\frac{1}{2}\right)}^2\times \frac{1}{2}=\frac{3}{8}$ (Note that there are three cases BBG, BGB, GBB)
(iii) $P$ (1 boy + 2 girls)${=}^3{\text{C}}_2\times {\left(\frac{1}{2}\right)}^2\times {\left(\frac{1}{2}\right)}^2=\frac{3}{8}$
(iv) $P$ (3 girls)$=\frac{1}{8}$
Event ${}^{\prime }{A}^{\prime }$ is associated with (iii) & (iv). Hence $P\left(A\right)=\frac{1}{2}$
Event ${}^{\prime }{B}^{\prime }$ is associated with (ii) & (iii). Hence $P\left(B\right)=\frac{3}{4}$
Event ${}^{\prime }{C}^{\prime }$ is associated with (i) & (ii). Hence $P\left(C\right)=\frac{1}{2}$
$P(A\cap B)=P\left(iii\right)=\frac{3}{8}=P\left(A\right).P\left(B\right).$ Hence $A$ and $B$ are independent of each other
$P(A\cap C)=0\ne P\left(A\right).P\left(C\right).$ Hence $A,B,C$ are not independent