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How to Put Negative Binomial Formula on Calculator

Reviewed by Calculator Editorial Team

The negative binomial distribution is a statistical model used to describe the number of trials needed to achieve a given number of successes. This guide explains how to input the negative binomial formula into your calculator for accurate statistical analysis.

What is the Negative Binomial Distribution?

The negative binomial distribution is a discrete probability distribution that describes the number of trials needed to achieve a specified number of successes in repeated, independent Bernoulli trials. It's commonly used in quality control, reliability testing, and other fields where counting the number of trials until a certain number of successes occurs is important.

Negative Binomial Formula

The probability mass function for the negative binomial distribution is:

P(X = k) = C(k-1, r-1) * (pr) * ((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) = combination of (k-1) things taken (r-1) at a time

The negative binomial distribution differs from the binomial distribution in that it models the number of trials until a specified number of successes, rather than the number of successes in a fixed number of trials.

How to Input the Formula on Your Calculator

Most scientific calculators can compute the negative binomial distribution, but you may need to use the combination function and exponentiation functions separately. Here's how to do it:

Calculator Requirements

You'll need a calculator with the following functions:

  • Combination function (often labeled as "nCr" or "C")
  • Exponentiation function (often labeled as "^" or "y^x")
  • Basic arithmetic operations (+, -, *, /)

If your calculator doesn't have a built-in negative binomial function, you can calculate it manually using the formula above.

Step-by-Step Guide

  1. Identify the parameters

    Determine the values for r (number of successes), p (probability of success), and k (number of trials).

  2. Calculate the combination

    Use the combination function to calculate C(k-1, r-1).

  3. Calculate the probability terms

    Calculate pr and (1-p)k-r using the exponentiation function.

  4. Multiply the terms

    Multiply the combination result by pr and (1-p)k-r to get the probability.

Worked Example

Let's calculate the probability of needing 10 trials to achieve 5 successes with a success probability of 0.3.

Example Parameters

  • k = 10 (number of trials)
  • r = 5 (number of successes)
  • p = 0.3 (probability of success)

  1. Calculate the combination

    C(10-1, 5-1) = C(9, 4) = 126

  2. Calculate pr

    0.35 ≈ 0.00243

  3. Calculate (1-p)k-r

    0.75 ≈ 0.16807

  4. Multiply the terms

    126 * 0.00243 * 0.16807 ≈ 0.0414

The probability of needing exactly 10 trials to achieve 5 successes is approximately 4.14%.

Frequently Asked Questions

What is 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?

Use the negative binomial distribution when you're interested in the number of trials until a certain number of successes occurs, such as in quality control or reliability testing.

Can I calculate the negative binomial distribution on my calculator?

Yes, you can calculate it manually using the combination and exponentiation functions, or use a calculator with built-in statistical functions if available.

What are the parameters for the negative binomial distribution?

The key parameters are r (number of successes), p (probability of success), and k (number of trials).