Probability Question 362
Question: 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
Options:
A) ½
B) Greater than ½
C) Less than ½
D) $ Greaterthan{\scriptscriptstyle 3!/! _4} $
Show Answer
Answer:
Correct Answer: B
Solution:
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}. $
BETA
