Q79 / 160 · TS EAMCET 2020 (14 Sep Shift 2) · Mathematics

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 atleast 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 three independent events $A, B$ and $C$ $\begin{array}{r} P(A \cup B \cup C)=P(A)+P(B)+P(C)-P(A) P(B) \\ -P(A) P(C)-P(B) P(C)+P(A) P(B) P(C) \end{array}$ The correct option among the following is

  • Correct answer
    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

Solution

$P_1, P_2, P_3$ are three independent events Probability of happening at least one event is $1-\left[\left(\bar{P}_1\right) \cdot\left(\bar{P}_2\right)\left(\bar{P}_3\right)\right]=1-\left(1-\bar{P}_1\right)\left(1-\bar{P}_2\right)\left(1-\bar{P}_3\right)$ $\therefore A$ is true. If there independent events are $A, B$ and $C$ $\begin{array}{r} \therefore P(A \cup B \cup C)=P(A)+P(B)+P(C)-P(A) P(B) \\ -P(A) P(C)-P(B) P(C)+P(A) P(B) P(C) \end{array}$ $R$ is also true. $\therefore \mathrm{A}$ is true and $\mathrm{R}$ is true and $\mathrm{R}$ is the correct explanation for $A$.

Hence, the correct answer is option: 'A'.

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