Picking numbers at random I pick two numbers uniformly at random from {1; 2; : : : ; 6n}, with replacement.

(a) What is the probability that both numbers choose are divisible by 2?
(b) What is the probability that both numbers choose are divisible by 3?
(c) What is the probability that either (a) or (b), or both, occurs?

As n approaches innity the answer to (c) approaches some limit. This might be said to be the probability that two integers chosen at random have a common factor or 2 or 3. It can be shown that, in this sense, the probability that two integers chosen at random have a common factor greater than 1 is 6/pi^2