Geometric Distribution Calculator
Compute exact and cumulative probabilities for the geometric distribution, plus mean, variance, and standard deviation.
For X (trial of first success): P(X = k) = (1 − p)k−1p, and P(X ≤ k) = 1 − (1 − p)k.
For Y (failures before first success): P(Y = k) = (1 − p)kp, and P(Y ≤ k) = 1 − (1 − p)k+1.
What is a geometric distribution?
The geometric distribution models the number of attempts needed to get the first success in repeated independent trials with a constant success probability p. If every trial has the same chance of success and each trial is independent, this is usually the right model.
Common examples include:
- How many sales calls until the first sale.
- How many coin flips until the first heads.
- How many manufactured items are tested before finding the first defect (if defect probability is constant).
How this geometric distribution calculator works
Step 1: Pick a definition
There are two standard versions of the geometric random variable:
- X = trial number of first success: values are 1, 2, 3, ...
- Y = failures before first success: values are 0, 1, 2, ...
Both are valid, but they index outcomes differently. This calculator supports both.
Step 2: Enter p and k
Enter a success probability p between 0 and 1, then enter your target integer k. The calculator returns:
- Exact probability at k
- CDF at k (probability up to k)
- Tail probabilities (at least / greater than)
- Mean, variance, and standard deviation
Why geometric distribution is useful
The geometric distribution is one of the cleanest models for “waiting time until first success.” It is easy to compute and interpret, and it has the memoryless property: the probability of needing more future trials does not depend on how many failures already happened.
Memoryless property
For the trial-number version, this means: P(X > s + t | X > s) = P(X > t). In plain language, if you already failed for 5 attempts, your future waiting time behaves like a fresh start.
Quick interpretation guide
- Large p (e.g., 0.7): success tends to happen quickly; mass is near small k.
- Small p (e.g., 0.05): longer waits are more likely.
- Mean for X is 1/p, so p = 0.2 implies about 5 trials on average to first success.
- Mean for Y is (1−p)/p, so p = 0.2 implies about 4 failures before first success.
Geometric vs. related distributions
Geometric vs. Binomial
Use geometric when you care about when the first success occurs. Use binomial when you care about how many successes in a fixed number of trials.
Geometric vs. Negative Binomial
Geometric is a special case of the negative binomial with target successes = 1. If you need waiting time until the 2nd, 3rd, or r-th success, use negative binomial.
Common mistakes to avoid
- Mixing up the two definitions (starting at 0 vs. starting at 1).
- Entering k as a non-integer.
- Using p outside the interval (0, 1).
- Applying geometric when trials are not independent or p changes over time.
Example use case
Suppose a support team closes a ticket on first response with probability p = 0.25. If X is the response number of first close, then:
- P(X = 3) = (0.75)2(0.25) = 0.140625
- P(X ≤ 3) = 1 − (0.75)3 = 0.578125
So there is about a 57.8% chance the first close happens by the third response.
Final note
This geometric distribution calculator is best for quick decision support, probability checks, and learning. If your scenario has changing probabilities or dependencies between trials, consider a different model.