question_answer

Let A, B, C be the events. If the probability of occurring exactly one event out of A and B is 1-a. out of B and C and A is 1-a and that of occurring three events simultaneously is \[{{a}^{2}}\], then the probability that at least one out of A, B, C will occur is

A) ½

B) Greater than ½

C) Less than ½

D) \[Greater\text{ }than\,\,{\scriptscriptstyle 3\!/\!{ }_4}\]

Correct Answer: B

Solution :

[b] P(exactly one event of A and B occurs)

\[=P[(A\cap B')\cup (A'\cap B)]\]

\[=P(A\cup B)-P(A\cap B)\]

\[=P(A)+P(B)-2P(A\cap B)\]

\[\therefore P(A)+P(B)-2P(A\cap B)=1-a\]             ? (1)

Similarly, \[P(B)+P(C)-2P(B\cap C)=1-2a\]? (2)

\[P(C)+P(A)-2P(C\cap A)=1-a\]               ? (3)

\[P(A\cap B\cap C)={{a}^{2}}\]

Now \[P(A\cup B\cup C)\]

\[=P(A)+P(B)+P(C)-P(A\cap B)-P(B\cap C)\]

\[-P(C\cap A)+P(A\cap B\cap C)\]

\[=\frac{1}{2}[P(A)+P(B)-2P(B\cap C)+P(B)+P(C)\]

\[-2P(B\cap C)+P(C)+P(A)-2P(C\cap A)]\]

\[+P(A\cap B\cap C)\]

\[=\frac{1}{2}[1-a+1-2a+1-a]+{{a}^{2}}\]

[using (1), (2), (3) and (4)]

\[=\frac{3}{2}-2a+{{a}^{2}}=\frac{1}{2}+{{(a-1)}^{2}}>\frac{1}{2}.\]