Often one would like to model a situation where you have one, and only one, probability assigned to a specific event, such as the probability of a machine failing. In these situations, there is a probability of success (p) and the probability of failure (1 - p). In a number of trials (x), one might have a certain number of successes (r), with the rest being failures. Maybe you are a manufacturer who will offer to replace a machine purchased by one of your customers if the machine breaks down a certain number of times in a number of uses as opposed to just repairing it. You could model this situation with a negative-binomial distribution. Assumptions surrounding the negative-binomial are
Under these conditions, the probability of observing the rth success on the xth trial is represented by the probability density function (pdf) P(X = x) = x-1Cr-1pr(1 - p)x - r.
The mean of such a distribution is r/p. The variance is r(1 - p) / p2. You can interpret the mean of this distribution as the expected number of trials needed before observing the rth success.
You can use the calculator below to calculate probabilities under your own scenario using the assumptions you input. The probability density function yields probabilities for a specified number of trials needed to obtain r successes, whereas the cumulative density function yields probabilities for each and every number of trials up to a specified number needed to obtain r successes. The graphs generated are shaped in accordance with the assumptions you enter.
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