Negative Binomial Calculator

A negative binomial experiment is a statistical experiment that has the following properties:

• The experiment consists of n repeated trials.
• Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
• The probability of success, denoted by p, is the same on every trial.
• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.
• The experiment continues until r successes are observed, where r is specified in advance.

Consider the following statistical experiment. You flip a coin repeatedly and count the number of times the coin lands on heads. You continue flipping the coin until it has landed 5 times on heads. This is a negative binomial experiment because:

• The experiment consists of repeated trials. We flip a coin repeatedly until it has landed 5 times on heads.
• Each trial can result in just two possible outcomes - heads or tails.
• The probability of success is constant - 0.5 on every trial.
• The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.
• The experiment continues until a fixed number of successes have occurred; in this case, 5 heads.

The probability distribution of a negative binomial random variable is called a negative binomial distribution. The negative binomial distribution is also known as the Pascal distribution.

Suppose we flip a coin repeatedly and count the number of heads (successes). If we continue flipping the coin until it has landed 2 times on heads, we are conducting a negative binomial experiment. The negative binomial random variable is the number of coin flips required to achieve 2 heads. In this example, the number of coin flips is a random variable that can take on any integer value between 2 and plus infinity. The negative binomial probability distribution for this example is presented below.

The number of trials refers to the number of attempts in a negative binomial experiment.

Suppose that we conduct the following negative binomial experiment. We flip a coin and count the number of flips until the coin has landed three times on Heads. If we need to flip the coin 5 times until the coin has landed on Heads 3 times, then 5 is the number of trials.

Each trial in a negative binomial experiment can have one of two outcomes. The experimenter classifies one outcome as a success; and the other, as a failure. The experiment continues until a specified number of successes is observed. The number of successes in a binomial experient refers to the number of successes required to terminate the binomial experiment.

Suppose that we conduct the following negative binomial experiment. We flip a coin and count the number of flips until the coin has landed three times on Heads. In this experiment, a coin flip that landed on heads would be considered a success. And the number of successes required to terminate the experiment would be three.

In a negative binomial experiment, the probability of success on any individual trial is constant. For example, the probability of getting Heads on a single coin flip is always 0.50. If "getting Heads" is defined as success, the probability of success on a single trial would be 0.50.

The negative binomial probability refers to the probability that a negative binomial experiment results in r - 1 successes after trial x - 1 and r successes after trial x.

For example, suppose we conduct a negative binomial experiment to count the number of coin flips required for a coin to land 2 times on Heads. We might ask: What is the probability that this experiment will require 5 coin flips? In this example, we would be asking about a negative binomial probability. (If you use the Negative Binomial Calculator to analyze this experiment, you will find that the probability that this experiment would require 5 coin flips is 0.125.)

Note: Some references define negative binomial probability differently. For a discussion of alternative definitions of negative binomial probability, see Stat Trek's negative binomial distribution tutorial.

A binomial experiment and a negative binomial experiment have exactly the same properties, except for one thing.

With a binomial experiment, we are concerned with finding the probability of r successes in x trials, where x is fixed. With a negative binomial experiment, we are concerned with finding the probability that the rth success occurs on the xth trial, where r is fixed.

The geometric distribution is a special case of the negative binomial distribution. It deals with the number of trials required for a single success. Thus, the geometric distribution is negative binomial distribution where the number of successes (r) is equal to 1.

With a negative binomial distribution, we are concerned with finding the probability that the rth success occurs on the xth trial, where r is fixed. With a geometric distribution, we are concerned with finding the probability that the first success occurs on the xth trial.

Of course! The geometric distribution is just a special case of the negative binomial distribution (see above question); so geometric distribution problems can be solved with the Negative Binomial Calculator.

The mean (μ) and standard deviation (σ) of a binomial distribution are calculated using the number of trials (n) and the probability of success (P) on each trial.

The mean is the average number of trials expected to get n successes.

μ = n * P

And the standard deviation measures the variability in the number of trials needed to reach n successes.

σ = sqrt [ n * (1 - P) / P² ]