Two numbers $k_1$ and $k_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $i^{k_1} + i^{k_2}$, $(i = \sqrt{-1})$ is non-zero, equals:
Step-by-Step Solution
Key Concept: $i^k$ is periodic with period 4, taking values $\{1, i, -1, -i\}$ with equal probability $1/4$ each. The sum is zero when the two values are negatives of each other.
Each of $i^{k_1}$ and $i^{k_2}$ takes values $\{1,i,-1,-i\}$ with prob $1/4$ each. $P(\text{sum} = 0) = 4/16 = 1/4$. $P(\text{non-zero}) = 1 - 1/4 = 3/4$.
Correct Answer: $\frac{3}{4}$