Question

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

Answer 1

$P(A)=\frac{1}{2},$
$P(A∪B)=\frac{3}{5},$and $P(B)=ρ$
(i)A and B are mutually exclusive events, then
$A∩B=ϕ$
$⇒P(AϕB)=0$
$∴P(A∪B)=P(A)+P(B)-P(A∩B)$
$⇒\frac{3}{5}=\frac{1}{2}+ρ$
$⇒ρ=\frac{3}{5}-\frac{1}{2}$
$=\frac{6-5}{10}$
$=\frac{1}{10}$

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(ii) $A$ and $B$ are independent events.
$∴P(A∪B)=P(A)+P(B)-P(A).P(B)$
$⇒\frac{3}{5}=\frac{1}{2}+ρ-\frac{1}{2}×ρ$
$⇒\frac{3}{5}-\frac{1}{2}=\frac{1}{2}ρ$
$⇒\frac{1}{2}ρ=\frac{1}{10}$
$⇒ρ=\frac{1}{5}$