Calculator Negative Binomial Probability

The negative binomial distribution describes the number of trials needed to achieve a specified number of successes in repeated, independent Bernoulli trials. This calculator computes the probability of needing exactly k trials to get r successes, given a success probability p.

What is Negative Binomial Probability?

The negative binomial distribution is a discrete probability distribution that extends the geometric distribution. While the geometric distribution models the number of trials until the first success, the negative binomial distribution models the number of trials until the r-th success.

Key characteristics of the negative binomial distribution:

Models the number of trials until r successes occur
Each trial has the same probability of success p
Trials are independent
Used when the number of successes is fixed, and we want to find the number of trials needed

The negative binomial distribution is often used in quality control, reliability engineering, and other fields where the number of trials until a certain number of successes is important.

Negative Binomial Formula

The probability mass function of the negative binomial distribution is given by:

P(X = k) = C(k-1, r-1) × p^r × (1-p)^k-r

Where:

k = number of trials (k ≥ r)
r = number of successes
p = probability of success on an individual trial
C(k-1, r-1) = binomial coefficient, "k-1 choose r-1"

The binomial coefficient C(k-1, r-1) represents the number of ways to arrange r-1 successes in k-1 trials. The formula accounts for all possible sequences of trials that result in exactly r successes in k trials.

Note: The negative binomial distribution is sometimes parameterized differently, with parameters n and p where n is the number of successes and p is the probability of success. This is equivalent to our parameterization with r = n and k = trials.

How to Use the Calculator

To calculate the negative binomial probability:

Enter the number of trials (k) - this must be greater than or equal to the number of successes (r)
Enter the number of successes (r) - this must be a positive integer
Enter the probability of success (p) - this must be between 0 and 1
Click "Calculate" to compute the probability
View the result and chart showing the probability distribution

The calculator will display the probability of exactly k trials being needed to achieve r successes, given the success probability p. It also provides a chart showing the probability distribution for different values of k.

Worked Example

Suppose we want to find the probability of needing exactly 10 trials to get 5 successes, with a success probability of 0.3.

Using the formula:

P(X = 10) = C(9, 4) × (0.3)⁵ × (0.7)⁵

C(9, 4) = 126

P(X = 10) = 126 × 0.00243 × 0.16807 ≈ 0.0426 or 4.26%

So, there's approximately a 4.26% chance of needing exactly 10 trials to get 5 successes with a 30% chance of success on each trial.

Applications of Negative Binomial Probability

The negative binomial distribution has several practical applications:

Quality control: Estimating the number of defective items in a production run
Reliability engineering: Modeling system failures until a certain number of components fail
Medical research: Analyzing the number of patients needed to achieve a certain number of successes in clinical trials
Sports analytics: Predicting the number of games needed to achieve a certain number of wins
Financial modeling: Estimating the number of trades needed to achieve a certain number of profitable outcomes

In each case, the negative binomial distribution helps model scenarios where the number of trials until a certain number of successes is important.

FAQ

What's the difference between binomial and negative binomial distributions?: The binomial distribution models the number of successes in a fixed number of trials, while the negative binomial distribution models the number of trials needed to achieve a fixed number of successes.
When should I use the negative binomial distribution instead of the geometric distribution?: Use the negative binomial distribution when you need to model the number of trials until a specific number of successes (r > 1), and use the geometric distribution when you only care about the first success (r = 1).
What happens if the number of trials is less than the number of successes?: The probability is zero because it's impossible to have more successes than trials. The calculator will show this result.
Can the negative binomial distribution be used for continuous data?: No, the negative binomial distribution is specifically for discrete data representing counts of trials until a certain number of successes.
How does the negative binomial distribution relate to the Poisson distribution?: When the number of successes r is large and the probability of success p is small, the negative binomial distribution approximates a Poisson distribution with parameter λ = r × (1-p)/p.