Negative Binomial (Pascal) Distribution

Created:October 4, 2024

Last Updated:August 14, 2025

This calculator helps you compute the probabilities of a negative binomial distribution given the number of successes needed (r) and the probability of success (p). You can find the probability of achieving a certain number of failures before the r-th success. The distribution chart shows the probability mass function (PMF) of the negative binomial distribution.

Calculator

Parameters

Number of successes needed (r):

Probability of success (p):

Calculation Type:

Number of failures (x):

Interactive Distribution Chart

Enter parameters and calculate to view the distribution

Learn More

What is the Negative Binomial Distribution?

The negative binomial distribution, also known as the Pascal distribution, is a discrete probability distribution that models the number of failures in a sequence of independent Bernoulli trials before a specified number of successes occurs. Each trial has the same probability of success (p).

In other words, to achieve

r

successes with

k

failures requires a total of

k + r

trials. The first

k + r - 1

trials must contain exactly

r - 1

successes, which follows a binomial distribution, and the final trial must be a success with probability

p

Combining these probabilities gives us the negative binomial formula:

P(X = k) =\binom{k+r-1}{k}p^{r-1}(1-p)^k \cdot p = \binom{k+r-1}{k} p^r (1-p)^k

Where:

$r$ is the number of successes needed
$k$ is the number of failures before achieving $r$ successes
$p$ is the probability of success on each trial

Properties

Mean: $E(X) = \frac{r(1 - p)}{p}$
Variance: $Var(X) = \frac{r(1 - p)}{p ^ 2}$

When to Use Negative Binomial vs Binomial Distribution?

Use Negative Binomial When:

You have a fixed number of successes you want to achieve
You want to know how many failures you'll encounter
You continue until you reach your success target
You're planning resources needed to meet a quota

Use Binomial When:

You have a fixed number of trials
You want to know how many successes you'll get
You stop after a predetermined number of attempts
You're analyzing performance over a set period

Practical Examples:

📞 Sales Scenario - Negative Binomial

"I need to make 5 sales this week. My success rate is 20%. What's the probability of getting exactly 8 rejections before my 5th sale?"

→ Fixed successes (5), calculating P(X = 8 failures)

📞 Sales Scenario - Binomial

"I will make exactly 50 calls today. My success rate is 20%. What's the probability of making exactly 12 sales?"

→ Fixed trials (50), calculating P(X = 12 successes)

🏭 Quality Control - Negative Binomial

"We need 10 good components for assembly. With 90% quality rate, what's the probability of encountering exactly 3 defective items?"

→ Fixed good items needed (10), calculating P(X = 3 defects)

🏭 Quality Control - Binomial

"We will test exactly 100 components today. With 90% quality rate, what's the probability that exactly 88 will pass inspection?"

→ Fixed tests (100), calculating P(X = 88 passes)

🏥 Medical Research - Negative Binomial

"Clinical trial needs 50 qualified patients. If 15% qualify, what's the probability of screening exactly 280 people before finding our 50th qualified patient?"

→ Fixed qualified patients (50), calculating P(X = 280 screenings)

🏥 Medical Research - Binomial

"We will screen exactly 500 people. If 15% qualify, what's the probability that exactly 68 will be eligible for the study?"

→ Fixed screenings (500), calculating P(X = 68 qualified)

Key Decision Factor: Ask yourself - "Am I stopping after a fixed number of trials (Binomial), or continuing until I achieve a target number of successes (Negative Binomial)?"

How to Calculate Negative Binomial Distribution in R

library(tidyverse)
r <- 3    # number of successes needed
p <- 0.4  # probability of success on each trial

# P(X = )
prob_exact <- dnbinom(7, size = r, prob = p) # 0.06449725
print(str_glue("P(X = 7): {prob_exact}"))

# P(X <= 9)
prob_cumulative <- pnbinom(9, size = r, prob = p) # 0.9165567
print(str_glue("P(X <= 9): {prob_cumulative}"))

# mean and variance
mean <- r * (1-p)/p
variance <- r * (1-p)/(p^2)
print(str_glue("Mean: {mean}")) # 4.5
print(str_glue("Variance: {variance}")) # 11.25

# plot
x <- 0:20
pmf <- dnbinom(x, size = r, prob = p)
pmf_df <- data.frame(x = x, pmf = pmf)

ggplot(pmf_df, aes(x = x, y = pmf)) +
  geom_col() +
  labs(
    x = "Number of failures before r successes",
    y = "Probability",
    title = "Negative Binomial Distribution PMF"
  ) +
  theme_minimal()

How to Calculate Negative Binomial Distribution in Python

Python

import scipy.stats as stats
import numpy as np
import matplotlib.pyplot as plt

r = 3    # number of successes needed
p = 0.4  # probability of success on each trial

# P(X = 7)
prob_exact = stats.nbinom.pmf(7, r, p)
print(f"P(X = 7): {prob_exact:.4f}")

# P(X <= 9)
prob_cumulative = stats.nbinom.cdf(9, r, p)
print(f"P(X <= 9): {prob_cumulative:.4f}")

# mean and variance
mean = r/p
variance = r * (1-p)/(p**2)
print(f"Mean: {mean:.4f}")
print(f"Variance: {variance:.4f}")

# plot
x = np.arange(0, 21)
pmf = stats.nbinom.pmf(x, r, p)
plt.figure(figsize=(10, 6))
plt.vlines(x, 0, pmf, colors="b", lw=2)
plt.plot(x, pmf, "bo", ms=8)
plt.xlabel("Number of failures before r successes")
plt.ylabel("Probability")
plt.title("Negative Binomial Distribution PMF")
plt.grid(True)
plt.show()