Two number k_1$k_1$ and k_2$k_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of i^k_1 + i^k_2$i^{k_1} + i^{k_2}$, (i = sqrt-1)$(i = \sqrt{-1})$ is non-zero, equals (1) frac12$\frac{1}{2}$ (2) frac14$\frac{1}{4}$ (3) frac34$\frac{3}{4}$ (4) frac23$\frac{2}{3}$

### Core Logic Total apples = 18 (3 rotten, 15 good). Random variable X = \0, 1, 2\$X = \{0, 1, 2\}$ representing the number of rotten apples. ### Step 1: Probability Distribution P(X = 0) = frac^15C_2^18C_2 = frac105153$$P(X = 0) = \frac{^{15}C_2}{^{18}C_2} = \frac{105}{153}$$ P(X = 1) = frac^3C_1 times ^15C_1^18C_2 = frac45153$$P(X = 1) = \frac{^3C_1 \times ^{15}C_1}{^{18}C_2} = \frac{45}{153}$$ P(X = 2) = frac^3C_2^18C_2 = frac3153$$P(X = 2) = \frac{^3C_2}{^{18}C_2} = \frac{3}{153}$$ ### Step 2: Expectation E(X) = 0 times frac105153 + 1 times frac45153 + 2 times frac3153 = frac51153 = frac13$$E(X) = 0 \times \frac{105}{153} + 1 \times \frac{45}{153} + 2 \times \frac{3}{153} = \frac{51}{153} = \frac{1}{3}$$ ### Step 3: Variance E(X^2) = 0 times frac105153 + 1 times frac45153 + 4 times frac3153 = frac57153$$E(X^2) = 0 \times \frac{105}{153} + 1 \times \frac{45}{153} + 4 \times \frac{3}{153} = \frac{57}{153}$$ Var(X) = E(X^2) - (E(X))^2 = frac57153 - left(frac13right)^2 = frac57153 - frac17153 = frac40153$$Var(X) = E(X^2) - (E(X))^2 = \frac{57}{153} - \left(\frac{1}{3}\right)^2 = \frac{57}{153} - \frac{17}{153} = \frac{40}{153}$$ ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Probability