Geometric Distribution Calculator
For a repeated trial with a fixed success probability, find the chance the first success lands exactly on trial x, the chance it happens by trial x, and the expected number of trials. Enter the success probability and the trial number.
What this calculator does
The Geometric Distribution Calculator estimates your result instantly from the values you enter, using standard statistics formulas. For a repeated trial with a fixed success probability, find the chance the first success lands exactly on trial x, the chance it happens by trial x, and the expected number of trials. Enter the success probability and the trial number. Calculations run in your browser, so nothing you type is sent to a server, and the output is an estimate rather than a certified or official figure.
Result
About Geometric Distribution Calculator
The geometric distribution models how many independent trials it takes to get the first success. Given a success probability p and a trial number x, this calculator computes the exact probability that the first success occurs on trial x, P(X = x) = (1 − p)^(x − 1) · p, and the cumulative chance it has occurred by then, P(X ≤ x) = 1 − (1 − p)^x.
It also reports the mean, 1 / p, the expected number of trials until the first success. The probability must be between 0 and 1 and the trial number at least 1.
Formula & Mathematical Calculation
P(X = x) = f(Probability of success (p), Trial of first success (x))
Calculated according to standard stats formulas implemented in the calculator core.
Calculation Example
With success probability p = 0.3 and trial x = 3:
- P(X = 3) = (0.7)² × 0.3 = 0.49 × 0.3 = 0.147
- P(X ≤ 3) = 1 − 0.7³ = 1 − 0.343 = 0.657
- Mean = 1 / 0.3 ≈ 3.3333
So the first success falls on trial 3 with probability 0.147, and about 3.333 trials are expected on average.
Key Benefits
- Reports the exact-trial probability, the cumulative probability, and the mean 1/p in one calculation
- Uses the geometric formulas (1-p)^(x-1) p and 1 - (1-p)^x directly for the point and cumulative chances
- Shows the expected number of trials until the first success, a handy planning figure
- Requires a probability between 0 and 1 and a trial number of at least 1; free and in-browser
How to Use this Calculator
-
Enter the success probability
Type the chance of success on a single trial into Probability of success (p); a value between 0 and 1. -
Enter the trial number
Type which trial you are asking about into Trial of first success (x); at least 1. -
Calculate
Press Calculate to read P(X = x), P(X <= x), and the Mean.
Assumptions & Limitations
What the calculation assumes
- Every value you enter is treated as an exact, known amount.
- Any rate or interval you enter is assumed to hold for the whole period unless the form asks for it separately.
- Results use standard published formulas and account only for the values entered above.
- Rounding is applied for display; the underlying calculation uses your full precision.
- Results are returned in the same units you enter, unless the tool states the output unit.
What it does not cover
- Geometric Distribution Calculator produces a calculated result for reference and learning, not a certified or official measurement.
- Real-world outcomes differ when tolerances, rounding rules or local standards apply.
- Definitions and standards vary by region and change over time; verify the convention that applies to your case.
- Figures are only as accurate as the inputs you supply — check them before acting on the result.
- For work where the result carries real consequences, confirm it against an authoritative source.
Frequently Asked Questions
Free to use, no account needed. Calculations run in your browser, so nothing you type is sent to a server. Results are estimates — how we build and check our calculators.