Geometric Distribution Calculator
When does the first success come?
Result
How it is calculated
Enter the success probability of one trial as a percentage p (above 0 and up to 100) and the trial number k (a whole number, 1 or more). With q = 1 − p/100 the tool gives three results. P(X = k) = qk−1 × p/100 is the chance that the first success happens exactly on trial k. P(X ≤ k) = 1 − qk is the chance of at least one success within the first k trials. The expected number of trials to the first success is 100 / p. Probabilities are shown as percentages with 2 decimals. A p of 0 or below, above 100, or a k that is zero or fractional returns no result.
Worked example
Each attempt succeeds with p = 20%, so q = 0.8, and we ask about trial k = 5. Exactly on trial 5: 0.84 × 0.2 = 0.4096 × 0.2 = 0.08192, which the tool shows as 8.19%. At most 5 trials: 1 − 0.85 = 1 − 0.32768 = 0.67232, shown as 67.23%. Expected number of trials: 100 / 20 = 5.00. So you would typically need about five attempts, yet only about an 8% chance that the first success lands exactly on the fifth.
Common mistakes
- Typing p as a fraction. The field is in percent: 0.2 means 0.2%, not 20%, and the expected trials would show as 500.
- Confusing "exactly on trial k" with "within k trials". They are different outputs: 8.19% against 67.23% in the example.
- Counting failures instead of trials. Here k includes the successful trial, so k = 1 means success on the very first try.
- Applying the model to dependent attempts, or to a success rate that changes over time. The formulas need every trial to have the same, independent probability.
About this calculator
The geometric distribution models the number of independent trials needed to get the first success, when each trial has the same success probability p. The probability that success first happens exactly on trial k is P(X=k) = (1−p)^(k−1) × p, the probability it happens within the first k trials is 1 − (1−p)^k, and the expected (average) number of trials needed is 1/p.
It’s the standard model whenever you’re asking “how long until the first hit” under repeated, independent chances: quality-control engineers use it to estimate how many items get inspected before the first defect turns up, sales teams use it to model how many cold calls precede the first conversion, and reliability engineers use it for the number of test cycles before a first component failure. In sports analytics it explains things like the expected number of free throws before the first miss for a player with a known shooting percentage.
This calculator takes your success probability p and trial count k and returns all three quantities — the exact-trial probability, the cumulative probability, and the expected number of trials — in one step.
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