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Geometric Distribution Calculator
The geometric distribution describes the number of trials needed to get the first success when each trial succeeds independently with probability p: how many times you roll a die before the first six, how many customers a salesperson calls before the first sale. Enter p, the probability type and k, and the calculator uses P(X = k) = (1 − p)ᵏ⁻¹p and the closed form P(X ≤ k) = 1 − (1 − p)ᵏ.
The output includes the single-term calculation, the cumulative rule, the mean 1/p and variance (1 − p)/p², a bar chart and a probability table. Because the distribution has no upper limit, the table runs far enough to cover almost all of the probability.
This page counts trials including the successful one (support k = 1, 2, 3, …), the convention used by SciPy and most textbooks. If your course counts failures before the first success, subtract 1 from k. The geometric distribution is the discrete analogue of the exponential distribution and shares its memoryless property.
Result and step-by-step solution
X = number of trials up to and including the first success. Mean 1/p = 5, variance (1 − p)/p² = 20, SD = 4.4721
- FormulaP(X = k) = (1 − p)k − 1 × p, P(X ≤ k) = 1 − (1 − p)k
- Single term at k = 3P(X = 3) = 0.82 × 0.2 = 0.128
- Apply the ruleP(X ≤ 3) = 0.488
- Probability table
k P(X = k) P(X ≤ k) P(X ≥ k) 1 0.2 0.2 1 2 0.16 0.36 0.8 3 0.128 0.488 0.64 4 0.1024 0.5904 0.512 5 0.08192 0.67232 0.4096 6 0.06554 0.73786 0.32768 7 0.05243 0.79028 0.26214 8 0.04194 0.83223 0.20972 9 0.03355 0.86578 0.16777 10 0.02684 0.89263 0.13422 11 0.02147 0.9141 0.10737 12 0.01718 0.93128 0.0859 13 0.01374 0.94502 0.06872 14 0.011 0.95602 0.05498 15 0.0088 0.96482 0.04398 16 0.00704 0.97185 0.03518 17 0.00563 0.97748 0.02815 18 0.0045 0.98199 0.02252 19 0.0036 0.98559 0.01801 20 0.00288 0.98847 0.01441 21 0.00231 0.99078 0.01153
If your textbook counts failures before the first success (Y = X − 1), subtract 1 from each k.
Frequently asked questions
- What is the expected number of trials until the first success?
- 1/p. If a free throw succeeds 20% of the time, the first success takes 5 attempts on average.
- How do I find the probability the first success takes more than k trials?
- P(X > k) = (1 − p)ᵏ, the probability of k failures in a row. Choose P(X > x) in the calculator.
- Which version of the geometric distribution is this?
- The number of trials up to and including the first success, so k starts at 1. For the failures-before-success version, use k − 1.
- What does memoryless mean?
- Having already failed several times does not change the chance of success on the next trial or the expected number of further trials.
Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.