Geometric Distribution Calculator
The Geometric Calculator makes it easy to compute individual and cumulative geometric probabilities. For help, read the Frequently-Asked Questions or review the Sample Problems.
To learn more, read Stat Trek's lesson on the geometric probability distribution.
Frequently-Asked Questions
Instructions: To find the answer to a frequently-asked question, simply click on the question.
What is a geometric experiment?
A geometric experiment is a statistical experiment that has the following properties:
- The experiment consists of x 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 the first success is observed.
The point of a geometric experiment is to count how many trials are required to observe the first success. The number of trials is a geometric random variable.
What is the number of trials?
The number of trials is a geometric random variable. It refers to the number of repeated trials required to produce the first success in a geometric experiment.
An example of a geometric experiment would be tossing a coin until it lands on heads. In this experiment, each toss is an independent trial; and the number of tosses required to observe the first head would be the number of trials.
What is the probability of success?
A geometric experiment consists of a series of repeated trials. Each trial can result in one of two possible outcomes - a success or a failure.
The probability of success, denoted by P, defines the likelihood that an individual trial will produce a success rather than a failure. The probability of success is the same for every trial.
How do you calculate geometric probability?
The formula for geometric probability is given below.
Simple formulas also exist to compute the cumulative probability that (a) at most x trials are required to achieve the first success in a geometric experiment or (b) at least x trials are required to achieve the first success. Here are those formulas:
- At most x trials: P(X ≤ x) = 1 - (1 - P)x
- At least x trials: P(X ≥ x) = (1 - P)x-1
What is the mean and standard deviation of a geometric distribution?
If we define the mean of the geometric distribution as the average number of trials required to produce the first success in a geometric experiment, then the mean and standard deviation of the geometric probability distribution are equal to:
μ = 1 / P
σ = sqrt [ (1 - P) / P2 ]
where μ is the mean of the geometric distribution, σ is the standard deviation of the geometric distribution, and P is the probability of a success on any given trial.
Sample Problems
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Suppose you randomly select cards with replacement from an
ordinary deck of playing cards. What is the probability that the first black-suited card
(i.e., either a club or spade) will be third card selectd?
Solution:
We know the following:
- The number of trials is three.
- The number of successes in the population is 26. Here, we define a success as choosing a black card, and there are 26 black cards in an ordinary deck of 52 playing cards. So, the probability of success is 26/52 or 0.5.
Therefore, we plug those numbers into the Geometric Calculator and hit the Calculate button.
The calculator reports that the geometric probability is 0.125. That is the probability of selecting the first black card on the third trial.
The calculator also reports cumulative probabilities.
For example, the probability of selecting the first black card on the first or second trial is 0.875. That is, P(X < 3) = 0.875.