Negative Binomial Distribution

Distributions and Statistical Models

This calculator determines the probability of a specific number of failures occurring before a predetermined number of successes is achieved in a series of Bernoulli trials.

Practical Examples

Explore real-world scenarios to understand how the negative binomial distribution is applied.

Quality Control in Manufacturing

manufacturing

A manufacturer inspects items from a production line. The probability of an item being non-defective is 0.95. What is the probability of finding 3 defective items before finding 100 non-defective ones?

r: 100, p: 0.95, k: 3

Basketball Free Throws

sports

A basketball player makes free throws with a 70% success rate. What's the probability that she misses 5 shots before making 10 successful ones?

r: 10, p: 0.70, k: 5

Ecological Sampling

biology

An ecologist is searching for a rare species of orchid, with a 5% chance of finding one in any given quadrat. What is the probability of searching 50 empty quadrats before finding 3 orchids?

r: 3, p: 0.05, k: 50

Sales Call Success

sales

A salesperson has a 20% chance of closing a deal on any given call. What is the probability they get 15 rejections before closing 4 deals?

r: 4, p: 0.20, k: 15

Other Titles
Understanding the Negative Binomial Distribution: A Comprehensive Guide
Dive deep into the concepts, applications, and mathematics behind the negative binomial distribution.

What is the Negative Binomial Distribution?

  • Core Concepts
  • Key Parameters
  • Comparison with Binomial Distribution
The negative binomial distribution is a discrete probability distribution that models the number of failures (k) in a sequence of independent and identically distributed Bernoulli trials before a specified, non-random number of successes (r) occurs. Each trial has only two possible outcomes: success or failure, with the probability of success (p) remaining constant throughout the trials.
Core Concepts
Unlike the binomial distribution, which counts successes in a fixed number of trials, the negative binomial distribution counts failures until a fixed number of successes is reached. This makes it particularly useful for modeling 'waiting time' scenarios.
Key Parameters
The distribution is defined by two parameters: 'r' (the number of successes to be achieved) and 'p' (the probability of success on an individual trial). The random variable 'X' represents the number of failures observed before the r-th success.
Comparison with Binomial Distribution
The key difference lies in what is fixed and what is random. In a binomial experiment, the number of trials is fixed, and the number of successes is the random variable. In a negative binomial experiment, the number of successes is fixed, and the number of trials (or failures) is the random variable.

Step-by-Step Guide to Using the Calculator

  • Inputting Parameters
  • Interpreting the Results
  • Using the Reset and Example Features
This calculator simplifies the process of working with the negative binomial distribution. Follow these steps to get your results.
Inputting Parameters
  1. Number of Successes (r): Enter the target number of successes. This must be a positive integer.
  2. Probability of Success (p): Enter the probability of a single success. This must be a number between 0 and 1.
  3. Number of Failures (k): Enter the specific number of failures you are interested in. This must be a non-negative integer.
Interpreting the Results
The calculator provides several key metrics: P(X=k) is the probability of observing exactly 'k' failures; P(X≤k) is the cumulative probability of observing 'k' or fewer failures; P(X>k) is the probability of observing more than 'k' failures. It also calculates the mean, variance, and standard deviation of the distribution.
Using the Reset and Example Features
Click 'Reset' to clear all input fields and results. Use the 'Examples' section to load pre-filled scenarios, which helps in understanding the practical applications of the formula.

Real-World Applications of the Negative Binomial Distribution

  • Quality Control
  • Biology and Ecology
  • Business and Finance
The negative binomial distribution is not just a theoretical concept; it has numerous practical applications across various fields.
Quality Control
In manufacturing, it can be used to model the number of defective items that must be inspected before a certain number of non-defective items are found. This helps in setting up efficient inspection plans.
Biology and Ecology
Ecologists use it to model species abundance. For instance, counting the number of non-host plants ('failures') an insect must visit before finding a certain number of host plants ('successes').
Business and Finance
In sales, it can predict the number of unsuccessful calls a representative might have to make before achieving a target number of sales. In finance, it can be applied to model the number of losing trades before a certain number of profitable ones.

Common Misconceptions and Correct Methods

  • Confusing with Geometric Distribution
  • Assuming Constant Probability
  • Ignoring Independence of Trials
Understanding common pitfalls can help in applying the negative binomial distribution correctly.
Confusing with Geometric Distribution
A common mistake is to confuse it with the geometric distribution. The geometric distribution is a special case of the negative binomial distribution where the number of successes (r) is exactly 1. For r > 1, the negative binomial distribution is required.
Assuming Constant Probability
The model assumes that the probability of success 'p' is constant for every trial. In real-world scenarios, this might not always be true (e.g., a player's free-throw percentage might change with fatigue). It's crucial to validate this assumption.
Ignoring Independence of Trials
The trials must be independent. The outcome of one trial should not influence the outcome of another. If trials are dependent (e.g., drawing cards without replacement), other statistical models should be used.

Mathematical Derivation and Formula

  • The Probability Mass Function (PMF)
  • Calculating the Mean and Variance
  • A Worked Example
The probability of observing 'k' failures before the 'r'-th success is given by the Probability Mass Function (PMF).
The Probability Mass Function (PMF)
The formula is: P(X = k) = C(k + r - 1, k) p^r (1-p)^k. Here, C(n, k) is the binomial coefficient, calculated as n! / (k! * (n-k)!). It represents the number of ways to arrange the k failures among the k+r-1 trials (since the last trial must be a success).
Calculating the Mean and Variance
The expected number of failures (mean) is μ = (r (1-p)) / p. The variance, which measures the spread of the distribution, is σ² = (r (1-p)) / p².
A Worked Example
Let's say we want to find 2 successes (r=2) with a success probability of 0.25 (p=0.25), and we want to know the probability of having 3 failures (k=3) first. P(X=3) = C(3+2-1, 3) (0.25)^2 (0.75)^3 = C(4, 3) 0.0625 0.421875 = 4 * 0.026367... ≈ 0.1055.