1. Negative Binomial Experiment
The negative binomial experiment is almost the same as a binomial experiment with one difference: a binomial experiment has a fixed number of trials. If the following five conditions are true the experiment is binomial: Fixed number of n trials.
 Each trial is independent.
 Only two outcomes are possible (Success and Failure).
 Probability of success (p) for each trial is constant.
 A random variable Y= the number of successes.
 The number of trials, n is not fixed.
 A random variable Y= the number of trials needed to make r successes.
2. Negative Binomial Distribution
A negative binomial distribution (also called the Pascal Distribution) is a discrete probability distribution for random variables in a negative binomial experiment. It is used when there are exactly two mutually exclusive outcomes, labelled “success” and “failure”. The random variable is the number of repeated trials, X, that produce a certain number of successes, r. In other words, it’s the number of failures before a success. This is the main difference from the binomial distribution: with a regular binomial distribution, you’re looking at the number of successes. With a negative binomial distribution, it’s the number of failures that counts. A negative binomial experiment must meet the following requirements [1]:
 The experiment must consist of a sequence of independent trials.

 Each trial results in a success (S) or failure (F).

 The probability of success is the same from trial to trial.

 The experiment continues until you have r successes, where r is a positive integer (i.e., 1, 2, 3, …).
Why is it called Negative Binomial?
When you hear the term negative, you might think that a positive distribution is flipped over the xaxis, making it negative. However, the “negative” part of negative binomial actually stems from the fact that one facet of the binomial distribution is reversed: in a binomial experiment, you count the number of Successes in a fixed number of trials; in the above example, you’re counting how many aces you draw. In a negative binomial experiment, you’re counting the Failures, or how many cards it takes you to pick two aces.Formula
Probability: b*(x; r, P) = _{x1}C_{r1} * P^{r} * (1 – P)^{x – r} where x=number of trials r = Successes Mean: μ = r / P where r is the number of trials P=probability of success for any trialSolving Negative Binomial Experiment Problems
The probability mass function (pmf) for the negative binomial distribution is:Where: r is the number of successes and p = the probability of success. Example question 1: You are surveying people exiting from a polling booth and asking them if they voted independent. The probability (p) that a person voted independent is 20%. What is the probability that 15 people must be asked before you can find 5 people who voted independent? Step 1: Find p, r and X. We are given (in the question) that p = 20%(.2) and r = 5. The number of failures, X, is 15 – 5 = 10. Step 2: Insert those values from Step 1 into the formula:
Step 3: Solve. The first part (14 over 4) is a combination (use our combinations calculator to find 14 choose 4). 1001*.2^{5}*.8^{10} = 0.034. The probability you’ll have to ask 15 people to get 5 votes for independent is .034, or 3.4%. Example question 2: you are running a weight loss experiment and need 5 people willing to participate who have never tried a weight loss program before. If the probability that a randomly selected person has never tried a weight loss regimen is 0.2, what is the probability you must ask 15 people before you can find 5 people who have never tried losing weight? Solution: Place the following information into the formula:

 r = 5 (number of successes = people who have never tried losing weight)

 x = 10 (number of failures = people who have tried losing weight)

 p = 0.2 = probability of success.