Probability · Question 11

Two numbers k1 and k2 are randomly chosen from the set of natural numbers. Then, the probability that the value of ik1 + ik2, (i = -1) is non-zero, equals

1. 34
2. 12
3. 14
4. 23

Answer

(1) 34

Solution

ik depends only on the remainder of k on division by 4, and the four remainders are equally likely.

So (ik1, ik2) has 4 4 = 16 equally likely cases, each of ik1, ik2 \i, -1, -i, 1\.

ik1 + ik2 = 0 ik2 = -ik1: for each value of ik1 exactly one value of ik2 works 4 cases.

P(sum = 0) = 416 = 14

P(sum non-zero) = 1 - 14 = 34

Answer: (1)