KCET 2025 Math Paper

To enhance understanding of the statements, let's examine each one in detail:Statement (I): If events E and F are independent, the complements of these events, $E^{'}$ and $F^{'}$, are also independent. This statement is true.When two events, E and F , are independent, the occurrence of one does not affect the likelihood of the other occurring, mathematically expressed as $P(E \cap F)=P(E) \times P(F)$ Similarly, the complements of these events, $E^{'}$ and $F^{'}$, retain this independence property. Statement (II): Two mutually exclusive events cannot be independent if they have non-zero probabilities. This statement holds true.Mutually exclusive events are defined by $A \cap B=\emptyset$, which implies $P(A \cap B)=0$. For these events to be independent, it would require $P(A \cap B)=P(A) \times P(B)$. However, if $P(A) \neq 0$ and $P(B) \neq 0$, then $P(A \cap B) \neq 0$, contradicting the independence condition. Thus, two non-zero mutually exclusive events cannot be independent.Conclusively, Statement I is true and Statement II is also true.