Cumulative Probability Calculator

How to Calculate Attempts Needed for a Target Confidence

This calculator answers a different question than a plain "at least one" probability: instead of telling you the probability for a fixed number of attempts, it works backward and solves for the smallest whole number of attempts, n, that clears a target confidence level. Starting from the same complement formula used for "at least one," $P = 1-(1-p)^n$, solving for n and rounding up to a whole attempt gives: $$n = \left\lceil \frac{\ln(1 - \text{target})}{\ln(1 - p)} \right\rceil$$

Why the Achieved Confidence Isn't Exactly the Target

The exact number of attempts that would hit your target confidence precisely is almost always a fraction — for example, 21.85 attempts, not a clean whole number. Since you can't make 0.85 of an attempt, the formula rounds that fraction up to the next whole attempt (22, in this example), which means the cumulative probability at that attempt count comes out slightly above your target rather than landing on it exactly. This is expected and unavoidable with a whole-number attempt count — the table below shows exactly how much the achieved confidence clears the target at your computed answer.

Reading the Cumulative Probability Curve

The curve climbs quickly at first and then flattens out, because each additional attempt only shrinks the remaining "failure" probability by a factor of (1-p), not by a fixed amount. Early attempts make large jumps in cumulative probability; later attempts make smaller and smaller gains as the probability approaches (but never quite reaches) 100%. The milestone table shows this shape directly — probability at 1 attempt, roughly halfway to your answer, at your computed answer, and at double your computed answer, so you can see how much of the climb happens before you reach your target.

Worked Examples

1. 10% chance, 90% confidence. $p=0.1$, target $90\%$: $n = \lceil \ln(0.1)/\ln(0.9) \rceil = \lceil 21.85 \rceil = 22$ attempts, achieving about $90.15\%$.
2. 5% chance, 95% confidence. $p=0.05$, target $95\%$: $n = \lceil \ln(0.05)/\ln(0.95) \rceil = \lceil 58.4 \rceil = 59$ attempts, achieving about $95.14\%$.
3. 50% chance, 99% confidence. $p=0.5$, target $99\%$: $n = \lceil \ln(0.01)/\ln(0.5) \rceil = \lceil 6.64 \rceil = 7$ attempts, achieving about $99.22\%$.
4. 1% chance, 50% confidence. $p=0.01$, target $50\%$: $n = \lceil \ln(0.5)/\ln(0.99) \rceil = \lceil 68.97 \rceil = 69$ attempts, achieving about $50.01\%$ — a rare event still needs dozens of tries just to reach a coin-flip's worth of confidence.
5. 25% chance, 75% confidence. $p=0.25$, target $75\%$: $n = \lceil \ln(0.25)/\ln(0.75) \rceil = \lceil 4.82 \rceil = 5$ attempts, achieving about $76.27\%$.

Cumulative Probability Terms You Should Know

Cumulative Probability — the combined probability of an event happening across an entire series of attempts, growing as more attempts are added.

Target Confidence — the cumulative probability threshold you want to reach or exceed, expressed as a percentage.

Ceiling Function — a rounding rule that always rounds up to the next whole number, used here because a partial attempt isn't a real attempt.

Complement Rule — the probability of an event equals 1 minus the probability that it doesn't happen; the basis of both the "at least one" and "attempts needed" formulas.