Question

Let $A, B, C$ be three 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. $1 / 2$
• B. Greater than $1 / 2$
• C. Less than $1 / 2$
• D. Greater than $3 / 4$

Answer 1

Answer: (B)

$P$ (exactly one event out of $A$ and $B$ occurs)
$\left.=P\left[A \cap B'\right) \cup\left(A' \cap B\right)\right]$

$=P(A \cup B)-P(A \cap B)=P(A)+P(B)-2 P(A \cap B)$
$\therefore P(A)+P(B)-2 P(A \cap B)=1-a\,\,\,\,\,\,...(1)$
Similarly, $P(B)+P(C)-2 P(B \cap C)=1-2 a\,\,\,\,\,\,...(2)$
$P(C)+P(A)-2 P(C \cap A)=1-a\,\,\,\,\,\,...(3)$
$P(A \cap B \cap C)=a^{2}\,\,\,\,\,\,...(4)$
Now $P(A \cup B \cup C)$
$=P(A)+P(B)+P(C)-P(A \cap B)-P(B \cap C)$
$-P(C \cap A)+P(A \cap B \cap C)$
$=\frac{1}{2}[P(A)+P(B)-2 P(B \cap C)+P(B)+P(C)$
$-2 P(B \cap C)+P(C)+P(A)-2 P(C \cap A)]+P(A \cap B \cap C)$
$=\frac{1}{2}[1-a+1-2 a+1-a]+a^{2}\,\,\,\,\,\,\,\,\,$[using (1), (2), (3) and (4)]
$=\frac{3}{2}-2 a+a^{2}=\frac{1}{2}+(a-1)^{2}>\frac{1}{2}$