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Question

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


A

3p+2p22

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B

p+3p24

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C

p+3p22

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D

3p+2p24

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Solution

The correct option is A

3p+2p22


We know that
P(exactly one of A or B occurs)
=P(A)+P(B)2P(AB)
Therefore, P(A)+P(B)2P(AB)=p ---------------------- (1)
Similarly, P(B)+P(C)2P(BC)=p ------------------------ (2)
and P(C)+P(A)2P(CA)=p ------------------------------- (3)
Adding (1),(2) and (3) we get
2[P(A)+P(B)+P(C)P(AB)P(BC)P(CA)]=3pP(A)+P(B)+P(C)P(AB)P(BC)P(CA)]=3p2---------------(4)
We are also given that P(ABC)=p2-----------(5)
Now, P(at least one of A, B and C)
=P(A)+P(B)+P(C)P(AB)P(BC)P(CA)+P(ABC)=3p2+p2=3p+2p22


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