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 C or A occurs) = P(exactly one of the events B or C occurs) 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. $\frac { 3 p + 2 p ^ { 2 } } { 2 }$
• B. $\frac { p + 3 p ^ { 2 } } { 4 }$
• C. $\frac { p + 3 p ^ { 2 } } { 2 }$
• D.

Answer 1

Answer: (A)

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

Answer 2

P_A+ P_B+ = P_B + P_C = P_C + P_A = p

From first two, we get

P_A = P_C

From 2^nd and 3^rd relation, we get

P_B = P_A

∴ P_A = P_B = P_C = p/2

∴ P(AUBUC) = S₁ – S₂ + S₃

= $\frac { 3 } { 2 } p - 0 + p ^ { 2 } = \frac { 3 p + 2 p ^ { 2 } } { 2 }$