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