Statistics Definitions > 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:

1. Fixed number of n trials.
2. Each trial is independent.
3. Only two outcomes are possible (Success and Failure).
4. Probability of success (p) for each trial is constant.
5. A random variable Y= the number of successes.

Example: Take a standard deck of cards, shuffle them, and choose a card. Replace the card and repeat twenty times. Y is the number of aces you draw.

The negative binomial is similar to the binomial with two differences (specifically to numbers 1 and 5 in the list above):

• The number of trials, n is not fixed.
• A random variable Y= the number of trials needed to make r successes.

Example: Take a standard deck of cards, shuffle them, and choose a card. Replace the card and repeat until you have drawn two aces. Y is the number of draws needed to draw two aces. As the number of trials isn’t fixed (i.e. you stop when you draw the second ace), this makes it a negative binomial distribution.

Why is it called Negative Binomial?

When you hear the term negative, you might think that a positive distribution is flipped over the x-axis, 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.

The Negative Binomial Formula

Probability:
b*(x; r, P) = _x-1C_r-1 * 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 trial