Geometric probability is one of 7 distributions formulas in the probability section of this library, and it is used at ap · university level.
Why geometric probability works
For the first success to land on trial k, every one of the k minus 1 earlier trials must have failed and then trial k must succeed. The trials are independent, so those probabilities multiply, giving a failure probability raised to the number of failures and one factor of p at the end.
What each symbol means
$p$ is constant success probability and $k=1,2,\ldots$.
Geometric probability: when it holds
Trials are independent with two outcomes and constant $p$.
When it stops applying
This distribution assumes the trial count is not fixed and you may keep going indefinitely. If attempts are capped, say at five, then trial seven is not a possible outcome, the listed probabilities no longer add to 1, and the missing tail must be handled as a separate no-success case.
Geometric probability: a worked example
With $p=.25$, first success on trial 3 has probability $(.75)^2(.25)$.
The mistake to avoid
What people do: Raising the failure probability to k instead of k minus 1.
Why it goes wrong: With a success probability of 0.25, the first success on trial 3 has probability 0.140625. Using an exponent of 3 gives 0.10547, because it counts a failure on the very trial that was supposed to succeed.
Do this instead: Count the failures, not the trials: reaching trial k means exactly k minus 1 failures came before it.
Geometric probability: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $p$ is constant success probability and $k=1,2,\ldots$.
- Check the conditions before substituting. Trials are independent with two outcomes and constant $p$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Distributions
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where geometric probability comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about geometric probability
What is the average number of trials until the first success?
It is 1 divided by p, so a success probability of 0.25 gives a mean of 4 trials. That matches the intuition that a one-in-four event takes about four attempts on average.
How do I find the chance of needing more than k trials?
Raise the failure probability to the power k, since that is the chance the first k trials all fail. For p of 0.25 and k of 4 that is about 0.3164.
Does a run of failures make a success more likely soon?
No, this distribution is memoryless. After ten failures, the remaining wait has exactly the same distribution it had at the start, which is why the gambler's fallacy is a fallacy.
Why do some books start the count at zero?
They count failures before the first success instead of counting trials. That version has a mean of 1 minus p over p, one less than this one, so check which convention a formula uses.