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
$1-\left[{\bar{P}}_1\right)\cdot \left({\bar{P}}_2\right)\left({\bar{P}}_3\right)]=1-\left(1-{\bar{P}}_1\right)\left(1-{\bar{P}}_2\right)\left(1-{\bar{P}}_3\right)$
$\therefore A$ 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 $A$.