Question: Suppose that X has the binomial distribution with parameters n and p, and that Y has the negative binomial distribution with parameters r and p,

Suppose that X has the binomial distribution with parameters n and p, and that Y has the negative binomial distribution with parameters r and p, where r is a positive integer. Show that Pr(X < r) = Pr(Y > n − r) by showing that both the left side and the right side of this equation can be regarded as the probability of the same event in a sequence of Bernoulli trials with probability p of success.

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