Question

A and B are two events such that $P(A)=0.54,P(B)=0.69 \text{ and } P(A∩B)=0.35$. Find$(i)P(A∪B)(ii)P(A´∩B´)(iii)P(A∩B´)(iv)P(B∩A´)$

Answer 1

It is given that$P(A) = 0.54$, $P(B) = 0.69$, $P(A ∩ B) = 0.35$
(i) We know that$P (A U B) = P(A) + P(B) - P(A ∩ B)$
$∴P (A U B) = 0.54 + 0.69 - 0.35 = 0.88$

(ii) $A'∩ B' = (A U B)'$ [by De Morgan’s law]
$∴P(A'∩ B') = P(A U B)' = 1 - P(A U B) = 1 - 0.88 = 0.12$

(iii)$P(A ∩ B') = P(A) - P(A ∩ B) = 0.54 - 0.35 = 0.19$

(iv) We know that
$n(B∩A')=n(B)-n(A∩B)$
$⇒\dfrac{n(B∩A')}{n(S)}=\dfrac{n(B)n(s)-n(A∩B)}{n(S)}$
$∴P(B∩A')=P(B)-P(A∩B)$
$∴P(B∩A')=0.69-0.35=0.34$