Geometric Distribution Calculator

What the geometric distribution calculator does

The geometric distribution describes how long you wait for a first success in a run of independent attempts. This calculator works out the chance of a given number of failures before that first success, along with the average wait, the variance, and the standard deviation.

It fits any "keep trying until it works" situation, from rolling for a six to dialling until someone picks up. Below is how it works and what its numbers mean.

How to use it

1. Enter the number of failures that come before the first success.
2. Enter the probability of success on a single attempt, as a decimal between 0 and 1.
3. Press Calculate for the probability and the summary measures, or Reset to clear it.

What the geometric distribution models

The geometric distribution answers the question of how many tries it takes to get the first success, when each try is independent and has the same chance of succeeding. Unlike the binomial, which fixes the number of trials in advance and counts successes, the geometric lets the trials run on until success arrives, and counts the wait.

This calculator counts the number of failures before the first success. So if success comes on the very first attempt, that is zero failures; if it comes on the third, that is two failures first. Framing it as failures before success is one common version of the distribution, and it is worth being clear that the input is the count of misses, not the attempt number.

How the probability is worked out

For the first success to come after a certain number of failures, every one of those failures has to happen first, and then the success. Since the attempts are independent, you multiply their chances:

P = (1 minus p)^{(number of failures)} × p

The first part is the chance of that many failures in a row, each with probability one minus p. The final times p is the success that ends the run. Because more failures means multiplying by another fraction below 1 each time, longer waits are always less likely than shorter ones, so the single most likely outcome is success on the very first try.

The mean, variance, and what they tell you

The calculator also reports the mean, which is the average number of failures you would expect before a success. For this version of the distribution the mean is one minus p, divided by p. The rarer a success is, the longer the expected wait, which matches intuition: a one-in-ten event averages nine failures before it lands.

The variance and standard deviation measure how much that wait varies from run to run. Waiting times for rare events are not just long on average but also highly variable, so the standard deviation is large when the success chance is small. Together the three numbers describe both the typical wait and how unpredictable it is.

The memoryless property

The geometric distribution has a curious feature called memorylessness. If you have already had a string of failures, that streak does not change the chance of success on the next attempt. The distribution has no memory of what came before, so past failures do not bring you any closer to a success.

This is the mathematics behind the gambler's fallacy. After several tails in a row, it feels like heads is overdue, but each flip is independent and the odds are unchanged. A long unlucky streak is genuinely no reason to expect success sooner, which is exactly what the memoryless property says, and it is worth keeping in mind whenever waiting for a run of luck to turn.

A worked example

Suppose each attempt has a 20 percent chance of success, so p is 0.2, and you ask about 3 failures before the first success. The chance is 0.8³ times 0.2, which is about 0.102, or 10.2 percent.

The mean number of failures is (1 minus 0.2) divided by 0.2 = 4, so on average you would expect about four misses before a success at these odds. The variance and standard deviation come out large, around 20 and 4.5, a reminder that although four failures is typical, the actual wait swings widely from one run to the next.

Entering your values

Enter the number of failures as a whole number and the success probability as a decimal between 0 and 1. The calculator returns the probability of that many failures before the first success, plus the mean, variance, and standard deviation. Remember the count is failures before success, so zero means success on the first attempt.

Questions people ask

What is the geometric distribution?

The distribution of how long you wait for a first success in a run of independent attempts with a constant success chance. This calculator counts the failures before that success.

How is it different from the binomial?

The binomial fixes the number of trials and counts successes. The geometric lets trials run until the first success and counts how long that takes.

Does it count failures or the trial number?

This version counts the failures before the first success, so success on the first attempt is zero failures. The mean is one minus p, divided by p.

What does memoryless mean?

That past failures do not change the chance of success on the next attempt. A losing streak does not make success any more due, which is why the gambler's fallacy is a fallacy.

References

A quick note on where the methods here come from. The geometric distribution is set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering the geometric distribution.

1. NIST/SEMATECH e-Handbook of Statistical Methods (the geometric distribution). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (the geometric distribution). https://openstax.org/details/books/introductory-statistics-2e