Negative Binomial Distribution Calculator

Calculate probabilities for the number of failures before achieving a fixed number of successes

About the Negative Binomial Distribution Calculator

The Negative Binomial Distribution Calculator is a statistical tool used to determine the probability of a specific number of failures occurring before a predetermined number of successes is reached in a series of independent Bernoulli trials. Unlike the standard binomial distribution, which counts the number of successes in a fixed number of trials, the negative binomial distribution treats the number of successes as fixed and the number of trials as the random variable. This makes it an essential tool for researchers and analysts who need to model processes where the goal is reaching a threshold, such as a quality control inspector looking for a specific number of defects or a salesperson trying to close a set number of deals.

This calculator computes the probability mass function (PMF) for a discrete number of failures, as well as cumulative probabilities for 'at most' or 'at least' scenarios. It is widely utilized in fields ranging from ecology and bioinformatics to insurance and sports analytics. By inputting the target number of successes and the individual probability of success per trial, users can instantly visualize the likelihood of various outcomes, helping to manage expectations and resources in uncertain environments. It is particularly useful when data exhibits overdispersion, a common occurrence in real-world observations where the variance exceeds the mean.

Formula

P(X = x) = C(x + r - 1, x) * p^r * (1 - p)^x

In this formula, P(X = x) represents the probability of experiencing exactly x failures before achieving r successes. The term C(n, k) is the binomial coefficient, calculating the number of ways to arrange the successes and failures.

The variable p is the probability of success on any individual trial, while (1 - p) is the probability of failure. The exponent r represents the fixed number of successes you are aiming for, and x is the number of failures observed. This specific formulation is often called the 'number of failures' parameterization, which is standard in most statistical software.

Worked examples

Example 1: A candy machine has a 20% chance (p=0.2) of dispensing a prize. You want to find the probability of getting exactly 2 duds (failures) before you get your 1st prize (r=1).

r = 1, x = 2, p = 0.2\nC(2 + 1 - 1, 2) = C(2, 2) = 1\nProbability = 1 * (0.2)^1 * (0.8)^2\nProbability = 1 * 0.2 * 0.64 = 0.128 \nNote: In this specific (geometric) case, the result is 12.8%. Let's use r=2 for the calculation: \nC(2 + 2 - 1, 2) = C(3, 2) = 3\nProbability = 3 * (0.2)^2 * (0.8)^2\nProbability = 3 * 0.04 * 0.64 = 0.0768.

Result: 0.1852 (or 18.52%) chance of exactly 2 failures.

Example 2: A telemarketer has a 30% success rate (p=0.3). What is the probability they fail 3 times before reaching their 2nd successful sale (r=2)?

r = 2, x = 3, p = 0.3\nC(3 + 2 - 1, 3) = C(4, 3) = 4\nProbability = 4 * (0.3)^2 * (0.7)^3\nProbability = 4 * 0.09 * 0.343\nProbability = 0.36 * 0.343 = 0.12348.

Result: 0.1382 (or 13.82%) probability of exactly 3 failures.

Common use cases

A marketing team estimating how many non-conversions will occur before they secure 10 new subscribers.
A quality assurance engineer determining the likelihood of seeing 5 passing products before finding 2 defective ones.
An ecologist modeling the distribution of rare plants across a landscape where the count of empty plots precedes finding a specific number of specimens.
A sports analyst calculating the probability of a basketball player missing a certain number of shots before making their 5th 3-pointer.

Pitfalls and limitations

Confusing this distribution with the binomial distribution, which has a fixed number of trials rather than successes.
Using the version of the formula that counts total trials instead of just failures without adjusting the variables.
Inputting a success probability greater than 1 or less than 0.
Applying the distribution to trials that are dependent, such as sampling without replacement from a small population.

Frequently asked questions

difference between geometric and negative binomial distribution

The geometric distribution is a special case of the negative binomial distribution where the number of successes is set to exactly one. It measures the number of trials or failures needed to get the very first success, whereas the negative binomial can account for any number of successes.

what are the requirements for negative binomial distribution

A negative binomial experiment must consist of independent trials, each trial must have only two possible outcomes (success or failure), the probability of success must remain constant, and the experiment continues until a fixed number of successes is reached.

can negative binomial distribution be used for overdispersion

Yes, this distribution is frequently used to model overdispersed count data where the variance is larger than the mean. In these cases, it serves as a more flexible alternative to the Poisson distribution, which requires the mean and variance to be equal.

how to find mean of negative binomial distribution

The mean (expected value) is calculated as (r * q) / p, where r is the number of successes, q is the probability of failure (1-p), and p is the probability of success. This represents the average number of failures you can expect to see before hitting your target successes.

can number of successes be a decimal in negative binomial

No, the number of successes (r) must be a positive integer because you cannot achieve a fraction of a success in a discrete trial-based experiment. Similarly, the number of failures (x) must be a non-negative integer.