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Abstract N points are independently drawn from the uniform distribution on (0, 1). Denote by E(n|N; p), the event: There exists a subinterval of (0, 1) of length p that contains at least n out of the N points. We find the probability, P(n|N; p), of E(n|N; p) for n>N/2 in terms of simple tabulated quantities.
Joseph Naus (Tue,) studied this question.