Mathematics - Probability Question with Solution | TestHub

$\mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)=\frac{1}{2}\Rightarrow \frac{\mathrm{P}(\mathrm{B}\cap \mathrm{A})}{\mathrm{P}(\mathrm{A})}=\frac{1}{2}\Rightarrow \mathrm{P}(\mathrm{B}\cap \mathrm{A})=\frac{1}{8}$ $\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)=\frac{1}{4}\Rightarrow \frac{\mathrm{P}(\mathrm{A}\cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{1}{4}\Rightarrow \mathrm{P}(\mathrm{B})=\frac{1}{2}$ $\mathrm{P}(\mathrm{A}\cap \mathrm{B})=\frac{1}{8}=\mathrm{P}(\mathrm{A})\cdot \mathrm{P}(\mathrm{B})$ $\therefore$ Events A and B are independent. Now, $\mathrm{P}\left(\frac{{\mathrm{A}}^{\prime }}{\mathrm{B}}\right)=\frac{\mathrm{P}(\mathrm{A}\cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{\mathrm{P}(\mathrm{A})\mathrm{P}(\mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{3}{4}$ and $\mathrm{P}\left(\frac{{\mathrm{B}}^{\prime }}{{\mathrm{A}}^{\prime }}\right)=\frac{\mathrm{P}\left({\mathrm{B}}^{\prime }\cap {\mathrm{A}}^{\prime }\right)}{\mathrm{P}\left({\mathrm{A}}^{\prime }\right)}=\frac{\mathrm{P}\left({\mathrm{B}}^{\prime }\right)\mathrm{P}\left({\mathrm{A}}^{\prime }\right)}{\mathrm{P}\left({\mathrm{A}}^{\prime }\right)}=\frac{1}{2}$