Question

Solve
Q.4. If P (not A) = 0.7, P (B) = 0.7 and P (B/A) = 0.5, find P (A/B) and $P\left(A\cup B\right)$.

Answer 1

Dear student
$$\begin{gathered} \mathrm{P}\left(\mathrm{not}\mathrm{A}\right)=0.7 \\ \Rightarrow \mathrm{P}\left(\mathrm{A}\right)=1-\mathrm{P}\left(\mathrm{notA}\right)\left[\mathrm{as}\mathrm{P}\left(\mathrm{A}\right)+\mathrm{P}\left(\mathrm{notA}\right)=1\right] \\ \Rightarrow \mathrm{P}\left(\mathrm{A}\right)=1-0.7=0.3 \\ \mathrm{P}\left(\mathrm{B}\right)=0.7 \\ \mathrm{P}\left(\mathrm{B}|\mathrm{A}\right)=0.5 \\ \mathrm{We}\mathrm{know}\mathrm{that}: \\ \mathrm{P}\left(\mathrm{B}|\mathrm{A}\right)=\frac{\mathrm{P}\left(\mathrm{B}\cap \mathrm{A}\right)}{\mathrm{P}\left(\mathrm{A}\right)} \\ \Rightarrow \mathrm{P}\left(\mathrm{B}|\mathrm{A}\right)\times \mathrm{P}\left(\mathrm{A}\right)=\mathrm{P}\left(\mathrm{B}\cap \mathrm{A}\right) \\ \Rightarrow \mathrm{P}\left(\mathrm{B}\cap \mathrm{A}\right)=0.5\times 0.3=0.15 \\ \mathrm{To}\mathrm{find}: \\ \mathrm{P}\left(\mathrm{A}|\mathrm{B}\right)=\frac{\mathrm{P}\left(\mathrm{A}\cap \mathrm{B}\right)}{\mathrm{P}\left(\mathrm{B}\right)} \\ =\frac{0.15}{0.7} \\ =0.214 \\ \mathrm{and}\mathrm{we}\mathrm{know}\mathrm{that} \\ \mathrm{P}\left(\mathrm{A}\cup \mathrm{B}\right)=\mathrm{P}\left(\mathrm{A}\right)+\mathrm{P}\left(\mathrm{B}\right)-\mathrm{P}\left(\mathrm{A}\cap \mathrm{B}\right) \\ =0.3+0.7-0.15 \\ =0.85 \end{gathered}$$
Regards