Free · Probability and statistical models

Trial Count for the Required Success

Find the probability that the specified success first occurs on a chosen trial.

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For fair independent trials, the second success occurs on trial 3 with probability 2 × 0.5³ = 25%.

P(N=n) = C(n−1,r−1) p^r (1−p)^(n−r)

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Trials are independent with the same success probability; n counts all trials including the final success.
Success and trial counts are positive whole numbers, with r no greater than n. The retained trial cap is 171.
The zero and one probability endpoints follow the usual discrete distribution, including the empty-failure product when n equals r.

It describes the chance that trial n is the rth success, including r−1 earlier successes.

Translate the pinned captured helper to direct standalone arithmetic, preserving every input, selector and output dependency.