Expected Rolls to Target Calculator
How many tries should you expect before a given outcome occurs? Enter the chance of success on a single trial and a number of trials n. The calculator returns the expected number of trials until the first success, the variance of that wait, and the probability of succeeding at least once within n trials. This is the geometric distribution, the model for waiting for a die face, a rare item drop, or any repeated all-or-nothing attempt. Every figure is exact, derived from the single-trial probability with no estimation.
Geometric distribution formulas
Expected trials = 1 / p
Variance = (1 - p) / p^2
P(success within n) = 1 - (1 - p)^n
P(first success on trial n) = (1 - p)^(n-1) * p
The expectation counts the trial on which the first success occurs, so a probability p of 1/6 gives an expected 6 trials. The within-n probability is one minus the chance of failing every one of the n trials.
Worked examples
- Rolling a specific die face: p = 1/6, expected 6 rolls.
- The chance of rolling that face within 10 rolls is 1 minus (5/6)^10, about 83.85 percent.
- The variance for p = 1/6 is (5/6) / (1/36) = 30.
- A 1 percent drop chance (p = 0.01) takes on average 100 attempts.
- For p = 0.01 the chance of at least one drop in 100 tries is about 63.40 percent.
Expected rolls to target: frequently asked questions
What is the geometric distribution?
The geometric distribution models the number of independent trials needed to get the first success, where each trial succeeds with probability p. It answers questions like how many dice rolls you expect before rolling a six, or how many attempts before a rare drop appears.
What is the expected number of trials?
The expected number of trials until the first success is 1 divided by p. For a single die rolling a specific face, p is 1/6, so the expected number of rolls is 6. The expectation is the long-run average over many repetitions, not a guarantee for any single run.
What is the variance of the wait?
The variance of the number of trials until the first success is (1 minus p) divided by p squared. The spread is wide for rare events, so the actual number of trials can vary considerably around the expected value.
What is the chance of succeeding within n trials?
The probability of at least one success in n independent trials is 1 minus (1 minus p) to the power n. This rises toward 1 as n grows but never quite reaches certainty for any finite n when p is below 1.
What inputs are required?
Enter the single-trial success probability p between 0 and 1, and a number of trials n to evaluate the within-n probability. If you know the number of equally likely outcomes instead, set p to 1 divided by that count.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Geometric distribution.
- NIST Digital Library of Mathematical Functions: Discrete distributions reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.