A Binomial Tail Inequality for Successes

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.