Talk:Bertrand's ballot theorem

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I have seen formulations of the Ballot theeorem where a sequence of n iid non-negative integer valued random variables is considered. The probability that the sum of k of them is less than k, 1<=k<=n is then (1-Sn/n), where Sn is the sum of all n of them. Is this the same problem written differently, or a similar, but different problem?