Geometric Probability Distribution Calculator

What makes a problem geometric?

A geometric probability problem repeats independent trials until the first success occurs. OpenStax describes this in its exact lesson on the geometric distribution. The wording “until the first success” is the biggest clue that the geometric distribution may apply.

This calculator supports both common textbook definitions: X as the number of trials until the first success, and X as the number of failures before the first success. Those two definitions are closely related, but they use slightly different formulas and starting values.

Two geometric formulas students confuse

The difference is whether the successful trial is counted. If x counts all trials, the successful trial is included. If x counts failures only, the success happens after those failures.

Memoryless property in plain language

The geometric distribution is memoryless. If each trial is independent and the probability of success stays the same, previous failures do not change the probability of success on the next trial. This is useful in theory, but real-world problems should only use it when the independence assumption is reasonable.

Frequently asked questions

• What is a geometric probability distribution?
  A geometric distribution models the number of repeated independent trials needed to get the first success. Each trial has the same probability of success. A simple example is asking how many coin flips are needed until the first head, or how many attempts are needed until the first correct response.
• What is the difference between trials until success and failures before success?
  Trials until success includes the successful trial in the count, so the smallest possible value is 1. Failures before success does not include the successful trial, so the smallest possible value is 0. The formulas differ by one exponent, which is why the calculator asks which definition you are using.
• How is geometric probability different from binomial probability?
  Binomial probability uses a fixed number of trials and counts how many successes occur. Geometric probability does not fix the total number of trials ahead of time; it asks how long it takes to get the first success. If a problem says “in 12 trials,” think binomial. If it says “until the first success,” think geometric.
• What does the mean of a geometric distribution mean?
  If X is trials until the first success, the mean is 1/p. It represents the long-run average number of trials needed to get a success. If p = 0.20, the expected number of trials is 5. That does not guarantee success on the fifth trial; it is a long-run average.
• Can p be written as a percent?
  Yes, but the calculator expects a decimal. A 20% chance should be entered as 0.20. A common mistake is entering 20 instead of 0.20, which would make the probability invalid.