Q76 / 180 · KCET 2025 · Mathematics

Consider the following statements. Statement (I): If E and F are two independent events, then $E^{\prime}$ and $F^{\prime}$ are also independent. Statement (II): Two mutually exclusive events with non-zero probabilities of occurrence cannot be independent. Which of the following is correct?

  • A
    Statement (I) is true and statement (II) is false
  • B
    Statement (I) is false and statement (II) is true
  • Correct answer
    C
    Both the statements are true
  • D
    Both the statements are false

Solution

$E$ and $F$ are two independent events, then $E^{\prime}$ and $F^{\prime}$ are also independent. Statement $I$ true $\begin{aligned} & \mathrm{A} \cap \mathrm{~B}=\phi \Rightarrow \mathrm{P}(\mathrm{~A} \cap \mathrm{~B})=0 \ldots(1) \\ & \mathrm{P}(\mathrm{~A}) \neq 0 \ldots(2) \\ & \mathrm{P}(\mathrm{~B}) \neq 0 \ldots(3) \end{aligned}$ From eq(1), (2) and (3) $\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \neq \mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B}) \quad$ Statement II is true

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

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