Negative Binomial Calculator (Trials Until the r-th Success)

What this tool does

Where the binomial distribution counts successes in a fixed number of trials, the negative binomial turns it round: it counts the trials needed to reach a fixed number of successes. It answers questions such as how many sales calls to land five deals, or how many rolls before three sixes. With r = 1 it is the geometric distribution — the wait for the first success, with mean 1 ÷ p. On average you need r ÷ p trials, but the spread is wide: the chance of needing more than the average is substantial. Trials are assumed independent with a constant chance of success.

Formula

P(r-th success on trial k) = C(k − 1, r − 1) pʳ (1 − p)ᵏ⁻ʳ · P(within k) = 1 − Σⱼ₌₀^(r−1) C(k, j) pʲ (1 − p)ᵏ⁻ʲ · mean = r ÷ p

Variables

Worked example

Limitations

• The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
• The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.