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

Question

A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A &cap; B) = 0.35. Find (i) P(A &cup; B) (ii) P(A &cap; B ) (iii) P(A &cap; B ) (iv) P(B &cap; A )

Philoid · Accepted Answer

It is given that P(A) = 0.54, P(B) = 0.69, P(A ∩ B) = 0.35

(i) We know that P (A ∪ B) = P(A) + P(B) − P(A ∩ B)

⇒ P (A ∪ B) = 0.54 + 0.69 − 0.35 = 0.88

∴ P (A ∪ B) = 0.88

(ii) A′ ∩ B′ = (A ∪ B)′ [by De Morgan’s law]

So, P (A′ ∩ B′) = P(A ∪ B)′ = 1 − P(A ∪ B) = 1 − 0.88 = 0.12

∴ P (A’ ∩ B’) = 0.12

(iii) P (A ∩ B′) = P(A) − P(A ∩ B) = 0.54 − 0.35 = 0.19

∴ P (A ∩ B’) = 0.19

(iv) We know that: P (B ∩ A′) = P(B) − P(A ∩ B)

⇒ P (B ∩ A′) = 0.69 – 0.35

∴ P (B ∩ A′) = 0.34