Given A and B are two events

And, P(A) = 0.54 P(B) = 0.69 P(A ∩ B) = 0.35

By definition of P(A or B) under axiomatic approach we know that:

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

We have to find-

i) P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

= 0.54 + 0.69 – 0.35 = 0.88

ii) P(A’ ∩ B’) = P(A ∪ B)’ {using De Morgan’s Law}

⇒ P(A’ ∩ B’) = 1 – P(A ∪ B) = 1 – 0.88 = 0.12

iii) P(A ∩ B’) = This indicates only the part which is common with A and not B ⇒ This indicates only A.

P(only A) = P(A) – P(A ∩ B)

∴ P(A ∩ B’) = P(A) - P(A ∩ B) = 0.54 – 0.35 = 0.19

iv) P(A’ ∩ B) = This indicates only the part which is common with B and not A ⇒ This indicates only B.

P(only B) = P(B) – P(A ∩ B)

∴ P(A’ ∩ B) = P(B) – P(A ∩ B) = 0.69 – 0.35 = 0.34