Statistics & probability
Trials until first success
Find the chance that the first success occurs on a particular trial.
A clear answer. Here’s how.
Uses the geometric convention X=number of trials through the first success, starting at 1. P(X=k)=(1−p)^(k−1)p and P(X>k)=(1−p)^k. Independent trials and constant success probability are assumed; a zero success probability has no finite waiting-time mean and is excluded.
See a worked example
Worked example
A fixed demonstration, separate from the values you enter above.
- Success probability each trial (0–1)
- 0.25
- Trial number k, counting the successful trial
- 4
Example result
- P(first success on trial k)
- 0.10546875
- P(first success by trial k)
- 0.68359375
- P(first success after trial k)
- 0.31640625
- Expected trials including success
- 4