geometric distribution

geometric distribution, in statistics, a discrete probability distribution that describes the chances of achieving success in a series of independent trials, each having two possible outcomes. The geometric distribution thus helps measure the probability of success after a given number of trials. It has numerous real-life applications.

In a geometric distribution, each event is called a Bernoulli trial. Each Bernoulli trial has two outcomes—success or failure—with a fixed probability. The probability of success in a trial does not change across trials. Each trial is thus independent of past or future trials. This makes a geometric distribution memoryless—in other words, an individual trial that is a failure will not improve or reduce the probability of the next trial being a success. If these conditions are not met, then a geometric distribution cannot be used to determine the probability of success.

A geometric distribution can be defined in two different ways:

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• by the number of trials required to achieve success (called a shifted geometric distribution)
• or by the number of failures before the success.

Geometric distribution formula

The formula to calculate the probability mass function (PMF)—or, simply, probability—of success after n trials is:

P(X = n) = (1 − p)^{(n − 1)} • pwhere p is the probability of success in an individual trial.

The expected value (EV), or the most likely outcome from repeating the trial in a geometric distribution, is:

E(X) = 1/p

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To instead calculate the probability mass function, or probability, of n failures before a successful trial, the above PMF formula is slightly modified as follows:

P(Y = n) = P(X = n + 1) = (1 − p)ⁿ • p

The expected value in this case is:E(Y) = E(X − 1) = (1 − p)/p

Because the geometric distribution is a discrete distribution (i.e., each trial can have only two outcomes), the PMF is used, as opposed to the probability density function (PDF), which is used for continuous distributions, such as the normal distribution.

An example of the geometric distribution

In order to calculate the probability of a person rolling a 3 on the fifth roll (and not earlier) of a standard six-sided die, n = 5 (the fifth roll) and p = 1/6 (the probability of getting each number on a six-sided die). Therefore:

P(X = 5) = (1 − 1/6)^{(5 − 1)} • 1/6 = 0.08038

The probability of rolling a 3 on the fifth roll (and not before that) of a six-sided die is thus 0.08038. The expected value in this case is 6, which means that a person can be expected to roll a 3 on the sixth roll using the die:

E(X) = 1/1/6 = 6

A geometric distribution can also describe the probability of a person rolling a 3 for the first time in a sequence of rolls, as in the chart, which shows the probabilities from the first roll to the 25th.

To calculate the probability that a person will fail to roll a 3 on five rolls using a standard six-sided die and then will roll a 3 on the sixth attempt:

P(Y = 5) = (1 − 1/6)⁵ • 1/6 = 0.06698

The expected value in this case would be:

E(Y) = (1 − 1/6)/1/6 = 5

Similar distributions

The geometric distribution is similar to the binomial distribution in that each trial can have only two outcomes: success or failure, with the probability of success being the same across trials. But, the geometric distribution is concerned with only the first instance of success, whereas the binomial distribution seeks to determine the total number of successes over a fixed set of trials. In the example above, the probability of rolling a 3 on the fifth roll (and not earlier) using a standard six-sided die was calculated. A binomial distribution would instead calculate the probability of the number of times a person would roll a 3 if the person rolled the die a total of five times.

Likewise, the exponential distribution is the continuous version of the geometric distribution. The exponential distribution is a continuous frequency distribution and is also memoryless. It describes the time taken by a continuous process, occurring at an average rate, to change its state. The exponential distribution has many practical applications, such as predicting the service times of servers in a fast-food restaurant.

Real-life applications

As the preceding example shows, a common application of the geometric distribution is in games of chance with two outcomes (a rolled die is either a 3 or not a 3, and a coin toss is either heads or tails). The geometric distribution is also used in sports—using past averages such as a pitcher’s strikeout rate in baseball or a pass completion rate in football (soccer)—to determine probabilities of certain events in games.

The geometric distribution also has applications in manufacturing, where it is important to monitor and identify defective products. If the defect rate of a production process is known, it can be used to determine the probability of finding a defective product during the quality-control process. Identifying this probability, in turn, helps to determine (and, ideally, improve) quality control.

In business and finance, geometric distributions can be used in a cost-benefit analysis, which is intended to aid managers’ decision making. This application, however, can be more challenging than the others described here, because the probabilities associated with individual events are subject to multiple internal and external factors that do not necessarily stay constant over time.