In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs. For example, if we define a "1" as failure, all non-"1"s as successes, and we throw a die repeatedly until the third time “1” appears (r = three failures), then the probability distribution of the number of non-“1”s that had appeared will be a negative binomial.