Negative Binomial Mean and Variance Calculator

Enter the required success count r and the constant success probability p for each independent trial. The calculator returns the expected failure count and its variance immediately, using the standard failures-before-the-r-th-success convention. It is useful for coursework, reliability planning, repeated-attempt processes, and any model where trials continue until a fixed success target is met.

Choose the negative binomial convention

The phrase negative binomial distribution is used for two closely related random variables, so identifying the convention is the first important step. This calculator defines the random variable as the number of failures observed before the r-th success. Under that definition, a process can finish with zero failures when its first r trials all succeed. Some textbooks and software instead count the total number of trials required to reach the r-th success. That total-trials variable is exactly r larger than the failure-count variable, so its mean is larger by r while its variance is unchanged. Enter r as a positive whole number representing the success target. Enter p as the probability that any one trial succeeds, written as a decimal greater than zero and no greater than one. For example, use 0.4 for a 40 percent success probability. The model assumes independent trials and the same p on every trial. If attempts become easier, harder, or dependent on earlier outcomes, these formulas no longer describe the process reliably. The calculator rejects zero because a positive success target can never be reached when success is impossible, leaving no finite mean or variance to return.

Understand how the moments are calculated

For the failure-count convention, the mean is r multiplied by one minus p, then divided by p. The variance is r multiplied by one minus p, then divided by p squared. The mean is the long-run average number of failures across many repetitions of the full process; it does not predict the exact outcome of one run. The variance describes dispersion around that mean in squared failure-count units. As p approaches one, successes become nearly certain, so both values approach zero. At p equal to one, every trial succeeds and the process reaches r successes without any failures, making both the mean and variance exactly zero. As p moves closer to zero, failures become more common and both moments grow quickly. Variance grows especially fast because its denominator contains p squared. These patterns are useful for checking an answer: neither result can be negative, variance is at least as large as the mean for valid inputs, and both values are zero at p equal to one. Floating-point arithmetic may produce long decimal expansions for some inputs. Preserve sufficient precision for later calculations and round only when displaying a final report, because early rounding can accumulate error in downstream formulas.

Apply the result to an appropriate process

Use this calculator when observations are independent Bernoulli trials that continue until a fixed number of successes occurs. A support team might model failed contact attempts before completing several successful contacts, a test laboratory might model rejected units before recording a target number of passing units, or a communications engineer might model unsuccessful transmissions before a required number of successful deliveries. In each case, r must be fixed in advance and p must remain stable from attempt to attempt. The returned mean helps estimate average failure volume, staffing, spare capacity, or expected waste. The variance helps describe uncertainty and can feed other analytic work, but it is not the same as a confidence interval or a probability of exceeding a threshold. This capability does not calculate point probabilities, cumulative probabilities, quantiles, or parameters estimated from observed data. It also does not test whether independence or constant probability is plausible. If successes are sampled without replacement from a small population, if attempts influence later attempts, or if different subjects have different success probabilities, another model may be more suitable. Treat the output as a consequence of the assumptions you provide, not as evidence that those assumptions hold.