Consider the following game. There are n people standing in a queue, and there are m n closed boxes, k of the boxes contain prizes. Each next person in the queue can open any box and if it contains a prize he'll receive that prize. One person can open only one box. What is the best position in the queue to maximize the probability of winning the prize?

I've already solved it partially by seeing that P(winning) for the 1st person is k/m, then finding that it is the same for the 2nd and for the 3rd, so it kinda has to be the same for everyone, but my teacher said this solution is not rigorous, so i need another one:(