Question

Choose the correct alternative in the following question:

$$\begin{gathered} \mathrm{Let}\mathrm{P}\left(A\right)=\frac{7}{13},\mathrm{P}\left(B\right)=\frac{9}{13}\mathrm{and}\mathrm{P}\left(A\cap B\right)=\frac{4}{13}.\mathrm{Then},\mathrm{P}\left(\overline{A}|B\right)= \\ \left(\mathrm{a}\right)\frac{5}{9}\left(\mathrm{b}\right)\frac{4}{9}\left(\mathrm{c}\right)\frac{4}{13}\left(\mathrm{d}\right)\frac{6}{13} \end{gathered}$$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{P}\left(A\right)=\frac{7}{13},\mathrm{P}\left(B\right)=\frac{9}{13}\mathrm{and}\mathrm{P}\left(A\cap B\right)=\frac{4}{13} \\ \mathrm{As},\mathrm{P}\left(\overline{A}\cap B\right)=\mathrm{P}\left(B\right)-\mathrm{P}\left(A\cap B\right) \\ =\frac{9}{13}-\frac{4}{13} \\ =\frac{5}{13} \\ \mathrm{Now}, \\ \mathrm{P}\left(\overline{A}|B\right)=\frac{\mathrm{P}\left(\overline{A}\cap B\right)}{\mathrm{P}\left(B\right)} \\ =\frac{\left({\displaystyle \frac{5}{13}}\right)}{\left({\displaystyle \frac{9}{13}}\right)} \\ =\frac{5}{9} \end{gathered}$$

Hence, the correct alternative is option (a).