A set contains n consecutive natural numbers,where n is an odd natural number. If two numbers are randomly picked from the set, what is the probability that the average of all the numbers that are left is the same as that of the n numbers?

The number of ways in which 2 numbers can be selected is nC2. For the average of the remaining numbers to be the same as that of the full set, the two numbers that have to be removed should be equidistant from the middle number of the set (since n is odd). This can happen in (n-1)/2 ways.

So, the required probability is (n-1)/2 / nC2 = 1/n