How this calculator works
A first success on trial k requires k - 1 failures followed by a success. The failures convention subtracts one from every trial count, so it changes the mean but not the variance. The waiting time is memoryless when trials are independent. A zero chance would make the waiting time infinite and is excluded.
Formula
Trials: P(T=k) = (1-p)^(k-1)p, E[T] = 1/p. Failures: F = T - 1. Var[T] = (1-p)/p².
Worked example
With p = 0.5, the first success on trial 3 has probability 0.125, and success by trial 3 has probability 0.875.
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Frequently asked questions
Which convention is the default?
Trials, beginning at one. Select failures to match software that starts at zero.
What happens when p = 1?
The first trial succeeds with certainty, or equivalently there are zero failures.
Can I use dependent trials?
No. Dependence changes the waiting-time probabilities.