Consider the following statements.Statement ( | vedclass

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(C) Statement $(I)$: If $E$ and $F$ are independent,then $P(E \cap F) = P(E)P(F)$. We know that $P(E^{\prime} \cap F^{\prime}) = P((E \cup F)^{\prime}

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Both the statements are true

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(C) Statement $(I)$: If $E$ and $F$ are independent,then $P(E \cap F) = P(E)P(F)$. We know that $P(E^{\prime} \cap F^{\prime}) = P((E \cup F)^{\prime}) = 1 - P(E \cup F) = 1 - [P(E) + P(F) - P(E \cap F)] = 1 - P(E) - P(F) + P(E)P(F) = (1 - P(E))(1 - P(F)) = P(E^{\prime})P(F^{\prime})$. Thus,$E^{\prime}$ and $F^{\prime}$ are independent. Statement $(I)$ is true.Statement $(II)$: If $E$ and $F$ are mutually exclusive,then $P(E \cap F) = 0$. For them to be independent,we require $P(E \cap F) = P(E)P(F)$. Since $P(E) > 0$ and $P(F) > 0$,$P(E)P(F) > 0$. Thus,$P(E \cap F) 
eq P(E)P(F)$,meaning they cannot be independent. Statement $(II)$ is true.

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?

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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?