Free 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
Expected Trials7.5000
Variance11.2500
P(X = k) vs Target Successes (r)
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.
Example Calculation
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
Frequently Asked Questions
Learn More
Understanding the Normal Distribution
Learn what the normal distribution is, why it matters in statistics, and how to use the bell curve for probability calculations, z-scores, and real-world data analysis.