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

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

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

Solution:

Given: $\mathrm{P}(\mathrm{A})=1 / 2, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=1 / 5$ and let the probability of event be $\mathrm{P}(\mathrm{B})=\mathrm{p}$

(i) Mutually exclusive

When A and B are mutually exclusive which implies that if two events cannot occur at the same moment, they are mutually exclusive or discontinuous.

Then $(A \cap B)=\varphi$

$\Rightarrow P(A \cap B)=0$

As we know that, $P(A \cup B)=P(A)+P(B)-P(A \cap B)$
Substituting the value and evaluating:

$\Rightarrow 3 / 5=1 / 2+p-0$

$\Rightarrow \mathrm{P}=3 / 5-1 / 2=1 / 10$

(ii) Independent

When A and B are independent.Two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other.

$\Rightarrow P(A \cap B)=P(A) \cdot P(B)$