Statistical Methods
Count failures and attempts consistently in a geometric model
Define the task
Decide whether the recorded endpoint is the number of failures before the first success or the total number of attempts through that success. The two variables differ by exactly one. In R the geometric distribution uses failures before success, so zero is a possible outcome. A statement such as “success on attempt four” means three preceding failures.
Carry out the check
The model assumes independent attempts with the same success probability p on each attempt. The probability of success within m attempts is 1−(1−p)^m. Mean failures are (1−p)/p and mean attempts are 1/p. Document a retry cap, because stopping after the cap produces incomplete waiting counts rather than an ordinary observed first-success time.
Worked hypothetical example
With a hypothetical probability of 0.25 per independent attempt, the mean number of failures is three and the mean number of attempts is four. The probability of success by the fourth attempt is 1−0.75^4, approximately 0.6836. Four attempts are an expectation, not a guarantee. Learning, fatigue or changing attempt conditions make a constant probability assumption questionable.
Method checked 11 November 2026 against R documentation: The Geometric Distribution.