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 occurring is –

• A.
• B.
• C. $\frac { p + 3 p ^ { 2 } } { 2 }$
• D. $\frac { 3 p + 2 p ^ { 2 } } { 4 }$

Answer 1

Answer: (A)

Answer 2

We know that P(exactly one of A or B occurs)

= P (1) + P (2) – 2P (A Ē B)

Therefore, P (1) + P (2) – 2P (A Ē B) = p … (i)

Similarly, P (2) + P (3) – 2P (B Ē C) = p … (ii)

and P (3) + P (1) – 2P (C Ē A) = p … (iii)

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

P (1) + P (2) + P (3) – P (A Ē B) – P (B Ē C) – P (C Ē A)

= $\frac { 3 \mathrm { p } } { 2 }$ … (iv)

We are also given that P (A Ē B Ē C) = p² … (v)

Now, P (at least one of A, B and C)

= P (1) + P (2) + P (3) – P (A Ē B) – P

(B Ē C) – P (C Ē A) + P (A Ē B Ē C)

= + p², [By (iv) and (v)]