Question

Mark the correct alternative in the following question:

$$\begin{gathered} \mathrm{If}A\mathrm{and}B\mathrm{are}\mathrm{two}\mathrm{events}\mathrm{such}\mathrm{that}\mathrm{P}\left(A\right)=0.4,\mathrm{P}\left(B\right)=0.3\mathrm{and}\mathrm{P}\left(A\cup B\right)=0.5,\mathrm{then}\mathrm{P}\left(\overline{B}\cap A\right)\mathrm{equals} \\ \left(\mathrm{a}\right)\frac{2}{3}\left(\mathrm{b}\right)\frac{1}{2}\left(\mathrm{c}\right)\frac{3}{10}\left(\mathrm{d}\right)\frac{1}{5} \end{gathered}$$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{P}\left(A\right)=0.4,\mathrm{P}\left(B\right)=0.3\mathrm{and}\mathrm{P}\left(A\cup B\right)=0.5 \\ \mathrm{As},\mathrm{P}\left(A\cup B\right)=0.5 \\ \Rightarrow \mathrm{P}\left(A\right)+\mathrm{P}\left(B\right)-\mathrm{P}\left(A\cap B\right)=0.5 \\ \Rightarrow 0.4+0.3-\mathrm{P}\left(A\cap B\right)=0.5 \\ \Rightarrow 0.7-\mathrm{P}\left(A\cap B\right)=0.5 \\ \Rightarrow \mathrm{P}\left(A\cap B\right)=0.7-0.5 \\ \Rightarrow \mathrm{P}\left(A\cap B\right)=0.2 \\ \mathrm{Now}, \\ \mathrm{P}\left(\overline{B}\cap A\right)=\mathrm{P}\left(A\right)-\mathrm{P}\left(A\cap B\right) \\ =0.4-0.2 \\ =0.2 \\ =\frac{2}{10} \\ =\frac{1}{5} \end{gathered}$$

Hence, the correct alternative is option (d).