A Binomial Tail Inequality for Successes

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I provide a monotonicity result on binomial tail probabilities in terms of the number of successes. Consider two binomial processes with n trials. For any k from 1 to n-1, as long as the expected number of successes in the first process is at least n(k-1)/(n-1) and the expected number of successes in the second process is at least k/(k-1) times larger than that of the first, then the probability of k-1 or fewer successes in the first process is strictly larger than the probability of k or fewer successes in the second.

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