We give a simple statistical proof of a binomial identity, by evaluating the Laplace transform of the maximum of n independent exponential random variables in two different ways. As a by product, we obtain a simple proof of an interesting result concerning the exponential distribution. The connections between a probabilistic approach and the statistical approach are discussed, which explains why certain binomial identities admit probabilistic interpretations. In the process, several new binomial identities are also obtained and discussed.