Result
Enter values and select Calculate.
Formula and method
For trials until the first success, P(X=k) = (1−p)ᵏ⁻¹p. For failures before the first success, P(Y=k) = (1−p)ᵏp. The engine uses log1p and expm1 forms for stable closed-form cumulative and survival probabilities.
Worked example
With p = 0.25 and the trials-until-success convention, P(X=3) = 0.140625, P(X≤3) = 0.578125, P(X≥3) = 0.5625, the mean is 4, and variance is 12.
Assumptions and boundaries
- The default convention counts trials, beginning at 1; the alternate convention counts failures, beginning at 0.
- Trials are independent and use a constant success probability.
- The success probability must be greater than 0 and at most 1.
- Binomial asks how many successes occur in fixed trials; geometric asks how long until the first success.
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