For the 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)=p2, where 0<p<1/2. Then the probability of at least one of the three events A, B and C occuring is
3p+2p22
We know that
P(exactly one of A or B occurs)
=P(A)+P(B)−2P(A∩B)
Therefore, P(A)+P(B)−2P(A∩B)=p ---------------------- (1)
Similarly, P(B)+P(C)−2P(B∩C)=p ------------------------ (2)
and P(C)+P(A)−2P(C∩A)=p ------------------------------- (3)
Adding (1),(2) and (3) we get
2[P(A)+P(B)+P(C)−P(A∩B)−P(B∩C)−P(C∩A)]=3p⇒P(A)+P(B)+P(C)−P(A∩B)−P(B∩C)−P(C∩A)]=3p2---------------(4)
We are also given that P(A∩B∩C)=p2-----------(5)
Now, P(at least one of A, B and C)
=P(A)+P(B)+P(C)−P(A∩B)−P(B∩C)−P(C∩A)+P(A∩B∩C)=3p2+p2=3p+2p22