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Question:

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)

Solution:

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) $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$

$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.8+0.5-0.32=0.98$

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