If P(A) = 0.8, P(B) = 0.5 and P(B|A) = 0.4, find
(i) P(A ∩ B) (ii) P(A|B) (iii) P(A ∪ B)
It is given that P(A) = 0.8, P(B) = 0.5, and P(B|A) = 0.4
(i) $P(B \mid A)=0.4$
$\therefore \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{A})}=0.4$
$\Rightarrow \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{0.8}=0.4$
$\Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.32$
(ii) $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}$
$\Rightarrow \mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{0.32}{0.5}=0.64$
(iii) $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.8+0.5-0.32=0.98$