NIMCET 2025 Paper

Concept:Compute probabilities from a finite sample space and check independence using the multiplication rule.Formula:$P(A) = \frac{\text{number of favorable outcomes}}{\text{total outcomes}}$; Events $A$ and $B$ are independent if $P(A \cap B) = P(A) \cdot P(B)$.Solution:Total outcomes = $4 \times 4 = 16$.Compute $P(A)$:$A = \{x \ge 2\}$ i.e. $x = 2,3,4$ with $y$ any of $1$ to $4$.Favorable outcomes = $3 \times 4 = 12$.$P(A) = \frac{12}{16} = \frac{3}{4}$. Option B is true.Compute $P(B)$:$B = \{y > x\}$.Count pairs: $x=1 ⇒ y=2,3,4$ (3); $x=2 ⇒ y=3,4$ (2); $x=3 ⇒ y=4$ (1); $x=4 ⇒$ none (0). Total = $6$.$P(B) = \frac{6}{16} = \frac{3}{8}$. Option D is true.Compute $P(A \cap B)$:Need $x \ge 2$ and $y > x$: $x=2 ⇒ y=3,4$ (2); $x=3 ⇒ y=4$ (1); $x=4 ⇒$ none (0). Total = $3$.Thus $P(A \cap B) = \frac{3}{16}$, but option A claims $\frac{1}{4} = \frac{4}{16}$. So option A is false.Check independence:$P(A) \cdot P(B) = \frac{3}{4} \cdot \frac{3}{8} = \frac{9}{32}$.$P(A \cap B) = \frac{3}{16} \neq \frac{9}{32}$, so $A$ and $B$ are not independent. Option C is true.Answer:Option A.