Zero Events Upper Limit Calculator (Rule of Three)
After n trials with no failures, accidents or side effects, how high the true rate can still be at your confidence level — and how many clean trials it takes to rule out a given rate.
Results
Upper limit for the true rate (exact)
4.870
%
Quick estimate (rule of three at 95 %)
5.000
%
That is at most 1 in
20.5
Trials with no events needed to rule that rate out
149
The dashed line marks the value you entered.
What this tool does
Seeing nothing in 60 trials does not mean the rate is zero: it means the true rate is probably below about 5 %, because a 5 % rate would produce no events in 60 trials only 1 time in 20. Solving (1 − p)ⁿ = 0.05 gives the exact limit, and for 95 % confidence it is close to 3 ÷ n — the famous rule of three. The reverse question is as useful: to be 95 % sure a rate is below 1 in 100 you need about 300 clean trials.
Formula
(1 − p)ⁿ = 1 − confidence ⟹ p_upper = 1 − (1 − confidence)^(1/n) ≈ 3 ÷ n at 95 %
Variables
| Symbol | Meaning | Unit |
|---|---|---|
n | Number of trials with no events at all | — |
cl | Confidence level | % |
tr | A rate you want to rule out | % |
UB | Upper limit for the true rate (exact) | % |
QK | Quick estimate (rule of three at 95 %) | % |
ONE | That is at most 1 in | — |
NT | Trials with no events needed to rule that rate out | — |
Worked example
- Number of trials with no events at all60
- Confidence level95 %
- A rate you want to rule out2 %
- Upper limit for the true rate (exact)4.870 %
- Quick estimate (rule of three at 95 %)5.000 %
- That is at most 1 in20.5
- Trials with no events needed to rule that rate out149
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Standard deviation and variance are calculated for a sample (dividing by n − 1). For a full population, divide by n instead.