For three events A, B and C, P (exactly one of the events A or B occurs) = P( exactly one of the events B or C occurs) = P (exactly one of the events C or A occurs) = p and P (all the three events occur simultaneously) = $p^2 $, where $ 0 < p < \frac{1}{2}$. Then the probability of atleast one if the three events A, B and C occuring is

$\frac{3p+2p^2}{2}$

We have,

$P(A) +P(B) -2P( A ∩ B) = p $ ...........(i)

$P(B) +P(C) -2P( B ∩ C) = p $ ...........(ii)

$P(C) +P(A) -2P( C ∩ A) = p $ ...........(iii)

and, $ P(A ∩ B ∩ C) = p^2 $ ..............(iv)

Adding (i), (ii) and (iii), we get

$2[P(A) +P(B) +P(C)+P(A ∩ B) - P (B  ∩ C) - P( A  ∩ C) ]= 3p $

$⇒ P(A) +P(B) +P(C) - P(A  ∩ B) - P(B  ∩ C) $

$P( A  ∩ C)=\frac{3p}{2}$

∴ Required probability

$ = P( A ∪ B ∪  C)$

$ = P(A) + P(B) + P(C) - P(A ∩ B) - P( B ∩C) - P( A ∩ C) + P( A ∩ B ∩C) $

$=\frac{3p}{2} + p^2 = \frac{3p+2p^2}{2}$