Given: A and B are two events.
P (A) = 0.54, P (B) = 0.69 and 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)
Now 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.
Hence 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.
Hence 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
