Geometric Distribution Calculator
Calculate the probability of when the first success occurs in repeated independent trials. Choose whether to count trials or failures, then explore exact, cumulative and interval probabilities with graphs and calculation steps.
Your geometric probability
Choose your convention and event, or try an example.
Show probability table
| Count (k) | P(X = k) | P(X ≤ k) | P(X ≥ k) |
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Show calculation
How to use the geometric distribution calculator
- Choose whether to count trials until the first success or failures before the first success.
- Enter the probability of success on each trial as a decimal or percentage.
- Select the probability event and enter a count or interval, or choose the target-probability calculation.
- Calculate to see the result, the probability distribution chart, summary statistics, and step-by-step solution.
Worked example: first success on the third trial
This example uses sample values, not your current inputs.
Suppose each independent attempt has a 25% chance of success. What is the probability that the first success occurs on the third attempt?
p = 0.25; q = 0.75; k = 3. Convention: trials until first success.
= 0.75² × 0.25
= 0.5625 × 0.25
= 0.140625
14.0625%
There is a 14.0625% chance that the first two attempts fail and the third attempt succeeds.
- Mean = 4 trials.
- Variance = 12.
- Standard deviation ≈ 3.4641 trials.
- P(X ≤ 3) = 0.578125.
- P(X ≥ 3) = 0.5625.
Geometric distribution formula and assumptions
The geometric distribution models the waiting time until the first success in repeated independent trials with a constant probability of success. Each trial has two possible outcomes: success or failure.
Trials until first success
P(X = k) = (1 − p)(k−1)p, k = 1, 2, 3, ...
Failures before first success
P(Y = k) = (1 − p)k p, k = 0, 1, 2, ...
Y = X − 1. Both definitions describe the same experiment but use different counts and means.
- p is the probability of success on each trial.
- 1−p is the probability of failure.
- Trials must be independent.
- p remains constant from trial to trial.
- The process stops at the first success.
- The distribution is discrete, with an infinite theoretical upper range.
Mean, variance, and standard deviation
When X counts trials until the first success, its mean is 1/p. When Y counts failures before the first success, its mean is (1−p)/p. Both conventions have variance (1−p)/p² and standard deviation √(1−p)/p.
Sources
- Penn State — Geometric and Negative Binomial Distributions ↗
- OpenStax — Geometric Distribution ↗
- Stat Trek — Geometric Distribution ↗
Reviewed: October 2026
Geometric PMF vs. CDF
The probability mass function (PMF) gives the chance of the first success occurring at an exact count. The cumulative distribution function (CDF) gives the chance that the first success has occurred by a specified count. The survival probability describes the chance of still waiting for success beyond a given count.
For trials until the first success, P(X = k) = (1−p)^(k−1)p, P(X ≤ k) = 1−(1−p)^k, and P(X > k) = (1−p)^k. These formulas answer different questions, so the inequality must match the intended event.
The memoryless property
The geometric distribution is memoryless. If several independent trials have already failed, the probability of success on the next trial is still p. Previous failures do not make a future success more likely or less likely when the success probability remains constant.
If each attempt succeeds with probability 0.25, then after five failures the next independent attempt still has a 25% chance of success. The probability of at least one success in the next three attempts is 1−0.75³ = 0.578125.
Geometric vs. binomial distribution
A binomial distribution counts the number of successes in a fixed number of independent trials. A geometric distribution counts how many trials or failures occur before the first success. Both use independent Bernoulli trials with a constant success probability, but they answer different questions.
For a fixed number of trials, use the Binomial Distribution Calculator.
Common questions
What is a geometric distribution?
A geometric distribution models how many independent trials are needed to obtain the first success when every trial has the same probability of success. It can count either all trials, including the successful one, or only the failures before that success.
What is the geometric distribution formula?
When X counts trials until the first success, P(X = k) = (1−p)^(k−1)p for k ≥ 1. Here p is the success probability on each trial. When X counts failures before the first success, the formula is P(X = k) = (1−p)^k p for k ≥ 0.
What is the difference between the two geometric distribution definitions?
The trials convention includes the successful attempt and starts at 1. The failures convention excludes the successful attempt and starts at 0. They describe the same experiment but differ by one count.
How do you calculate the probability of the first success on trial k?
Multiply the probability that the first k−1 trials fail by the probability that trial k succeeds: P(X = k) = (1−p)^(k−1)p.
How do you calculate cumulative geometric probability?
For trials until the first success, the probability of succeeding by trial k is P(X ≤ k) = 1−(1−p)^k. This includes success on any of the first k trials.
What is the difference between P(X ≥ k) and P(X > k)?
For trials until the first success, P(X ≥ k) means the first success occurs on trial k or later; P(X > k) means it occurs strictly after trial k. The formulas are (1−p)^(k−1) and (1−p)^k, respectively.
What are the mean and variance of a geometric distribution?
When counting trials until the first success, the mean is 1/p and the variance is (1−p)/p². When counting failures before success, the mean is (1−p)/p. The variance is the same in both conventions.
Can the probability of success be 100%?
Yes. If p = 1, every trial succeeds. The first success always occurs on trial 1, so the trials convention has X = 1 with certainty and the failures convention has zero failures with certainty.
What does the memoryless property mean?
It means that previous failures do not change the probability distribution of the remaining wait when trials are independent and the success probability stays constant. After any number of failures, the chance of success on the next attempt remains p.
How many attempts are needed for a 95% chance of success?
Use the target-probability calculation with your success probability p. The calculator finds the smallest whole-number trial count k for which 1−(1−p)^k is at least 0.95. For p = 0.3, the minimum is 9 trials.
What is the difference between geometric and binomial distributions?
The binomial distribution counts successes within a fixed number of trials. The geometric distribution counts the waiting time until the first success. Both assume independent trials with a constant success probability.
Is the geometric distribution continuous or discrete?
It is discrete because it describes whole-number counts of trials or failures. A related continuous waiting-time model is the exponential distribution.
Can I use this calculator when the probability changes between attempts?
No. The standard geometric distribution assumes a constant probability of success and independent trials. If probabilities change between attempts, these formulas generally do not apply directly.
Is a geometric distribution the same as geometric mean?
No. A geometric distribution is a probability model for waiting until the first success. A geometric mean is a type of average calculated from multiple values.
Use these tools for mathematical information, not as a substitute for checking assumptions. Read our methodology or disclaimer.