Question

Mother, father and son line up at random for a family picture. If A and B are two events given by A = Son on one end, B = Father in the middle, find P (A/B) and P (B/A).

Answer 1

Consider the given events.
A = Son standing on one end
B = Father standing in the middle

$$\begin{gathered} \mathrm{Clearly}, \\ S=\left\{MFS,MSF,FSM,FMS,SMF,SFM\right\} \\ A=\left\{MFS,FMS,SMF,SFM\right\}, \\ B=\left\{MFS,SFM\right\} \end{gathered}$$

$$\begin{gathered} \mathrm{Now}, \\ A\cap B=\left\{MFS,SFM\right\} \\ \left(\mathrm{i}\right)\mathrm{Required}\mathrm{probability}=P\left(A/B\right)=\frac{n\left(A\cap B\right)}{n\left(B\right)}=\frac{2}{2}=1 \\ \left(\mathrm{ii}\right)\mathrm{Required}\mathrm{probability}=P\left(B/A\right)=\frac{n\left(A\cap B\right)}{n\left(A\right)}=\frac{2}{4}=\frac{1}{2} \end{gathered}$$