Negative Binomial Distribution Calculator

Your details

The "failures" form asks how many failures occur before the r-th success. The "trials" form asks how many total trials are needed to achieve the r-th success. Both describe the same distribution - they are just different ways to count outcomes.
The target number of successes. Must be a positive integer. For example, r = 5 means you want to observe 5 successful outcomes.
The probability of success on any single trial. Must be strictly between 0 and 1. For example, p = 0.4 means each trial has a 40% chance of success.
The specific number of failures to evaluate. For example, k = 4 means you want P(exactly 4 failures before the r-th success).
Exact probability P(X = target)
0.092897

The probability mass function value at your specified outcome.

P(X < target)0.17367
P(X <= target)0.266568
P(X > target)0.733432
P(X >= target)0.82633
Mean (expected value)7.5
Variance18.75
Standard deviation4.3301
Mode5
P(X = target)0.092897
P(X <= target)0.266568
P(X > target)0.733432

P(X = 4) = 0.092897 for NB(r = 5, p = 0.4)

  • There is a 9.29% chance of observing exactly this outcome. The probability of achieving 5 successes on or before failure 4 is 26.66%.
  • On average, you expect 7.50 failures per 5 successes (with p = 0.4).
  • The most likely outcome (mode) is 5 failures. Your target is 1 failure below the mode.
  • Standard deviation is 4.3301, so roughly 95% of outcomes fall within about -1.2 to 16.2 failures.
  • More than half of outcomes exceed 4 failures (P(X > target) = 73.34%).

Next stepTo explore the full probability distribution, see the chart below. Switch to "Trials" mode to view total-trials results instead of failure counts.

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