Given that the events $A$ and $B$ are such that $P(A) = \cfrac{1}{2},P(A \cup B) = \cfrac{3}{5}$ and $P(B) = p.$ Find $p$ if they are (i) mutually exclusive (ii) independent.

(i) When $A$ and $B$ are mutually exclusive, then

$A \cap B = \phi \Rightarrow P(A \cap B) = 0 \Rightarrow P(A \cup B) = P(A) + P(B)$

$\Rightarrow \cfrac{3}{5} = \cfrac{1}{2} + p \Rightarrow p = \cfrac{3}{5} - \cfrac{1}{2} = \cfrac{{6 - 5}}{{10}} = \cfrac{1}{{10}}$

(ii) When $A$ and $B$ are independent, then $P(A \cap B) = P(A)P(B)$

$\Rightarrow P(A \cup B) = P(A) + P(B) - P(A)P(B)$

$\Rightarrow \cfrac{3}{5} - \cfrac{1}{2} = \cfrac{{2p - p}}{2} \Rightarrow \cfrac{p}{2} = \cfrac{{6 - 5}}{{10}} \Rightarrow p = \cfrac{2}{{10}} = \cfrac{1}{5}$