Given,
P (exactly one of A or B occurs) = p
P (Exactly one of B or C occurs) = p'
P (exactly one of C or A occurs) = p'
and p (All three occurs simultaneously) = p2.
i. e., P(A) + P(B) – 2P(A ∩ B) = p…(i)
P(B) + P(A) – 2P(B ∩ C) = p…(ii)
P(C) + P(A) – 2P(C ∩ A) = p …(iii)
and, P(A ∩ B ∩ C) = p2 …(iv)
(i) + (ii) + (iii)
⇒ P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(C ∩ A) = \(\frac{3}{2}p\)
We know,
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(A ∩ C) + P(A ∩ B ∩ C)
\(=\frac{3}{2}p + p^2\)
\(= \frac{3p + 2p^2}{2}\)
∴ The probability that at least one of the three events A, B and C occurs is \(\frac{3p + 2p^2}{2}\).