On maximal tail probability of sums of nonnegative, independent and identically distributed random variables

We consider the problem of finding the optimal upper bound for the tail probability of a sum of k nonnegative, independent and identically distributed random variables with given mean  x. For k=1 the answer is given by Markov's inequality and for k=2 the solution was found by Hoeffding and Shrikhande in 1955. We show that the solution for k=3 as well as for general k, provided x≤1/(2k−1), follows from recent results of extremal combinatorics.