Negative Binomial Calculator
Calculate the probability that the r-th success occurs on the k-th trial in a sequence of independent Bernoulli trials.
P(X = k)
0.10450944
P(X = k) vs Target Successes (r)
Formula
## How to Calculate Negative Binomial Probability ### Formula **P(X = k) = C(k-1, r-1) * p^r * (1-p)^(k-r)** The negative binomial distribution models the number of trials needed to achieve exactly r successes. The last trial must be a success (the r-th), and the preceding k-1 trials must contain exactly r-1 successes. The expected number of trials is r/p.
Exemplo Resolvido
A salesperson closes 40% of pitches. What is the probability that the 3rd sale happens on the 8th pitch?
- 01Need r=3 successes on trial k=8
- 02C(7,2) = 21
- 03p^r = 0.4^3 = 0.064
- 04(1-p)^(k-r) = 0.6^5 = 0.07776
- 05P(X=8) = 21 * 0.064 * 0.07776 ≈ 0.10450
- 06Expected trials = 3 / 0.4 = 7.5
Perguntas Frequentes
How does negative binomial relate to geometric?
The geometric distribution is the special case of the negative binomial with r=1 (waiting for the first success). The negative binomial sums r independent geometric random variables.
Why is it called "negative" binomial?
The name comes from the mathematical relationship to the binomial series with negative exponents. Alternatively, it can be viewed as "inverted" binomial: instead of fixing trials and counting successes, it fixes successes and counts trials.
What is the alternative parameterization?
Some texts define X as the number of failures before the r-th success, giving P(X=k) = C(k+r-1, k) * p^r * (1-p)^k. Our version counts total trials including the final success.
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