Question

Consider The following statements:
Assertion (A): If $P_{1}, P_{2}, P_{3}$ are probability of happening of three independent events, then probability of happening of at least one of them is $1-\left[\left(1-P_{1}\right)\left(1-P_{2}\right)\left(1-P_{3}\right)\right]$
Reason (R): For any tlnee independent events A, B and C $P(A \cup B \cup C)=P(A)+P(B)+P(C)-P(A) P(B)-P(A) P(C)-P(B) P(C)+P(A) P(B) P(C)$
The correct option among the following is

• A. (A) is true, (R) is true and (R) is the correct explanation for (A)
• B. (A) is true, (R) is true but (R) is not the correct explanation for (A)
• C. (A) is true but (R) is false
• D. (A) is false but (R) is true

Answer 1

Answer: (A)

$P_{1}, P_{2}, P_{3}$ are three independent events Probability of happening at least one event is
$\left.1-\left[\bar{P}_{1}\right)\cdot\left(\bar{P}_{2}\right)\left(\bar{P}_{3}\right)\right]=1-\left(1-\bar{P}_{1}\right)\left(1-\bar{P}_{2}\right)\left(1-\bar{P}_{3}\right)$
$\therefore A$ is true. If there independent events are $A, B$ and $C$
$\therefore P(A \cup B \cup C)=P(A)+P(B)+P(C)-P(A) P(B)$
$-P(A) P(C)-P(B) P(C)+P(A) P(B) P(C) R$ is also true.
$\therefore A$ is true and $R$ is true and $R$ is the correct explanation for $A$.