Hong Life CalculatorsUnderstanding the at least once probability calculator — concept and principle
An at-least-once probability calculator answers questions like "if an item drops with a 1% chance, what is my chance of getting it in 100 tries?" and "how many tries do I need for a 90% chance?". It is used for game drop rates and gacha pulls, quality-control sampling, repeated medical or safety risks, and any situation where the same independent chance is repeated.
How it is calculated
- P(at least once) = 1 − (1 − p)ⁿ, where p is the chance per try as a fraction (1% = 0.01) and n is the number of tries.
- Tries for a target probability T = ln(1 − T) ÷ ln(1 − p). The calculator shows this for T = 50%, 90% and 99%.
The tries are shown with decimals; in practice round up to the next whole number to reach the target. The chance per try must be greater than 0 and less than 100%.
The principle
"At least once" is the complement of "never". If each try fails with probability 1 − p and tries are independent, the probability of failing every one of n tries is (1 − p) multiplied by itself n times. Subtracting that from 1 gives the chance of at least one success. Solving (1 − p)ⁿ = 1 − T for n with logarithms gives the number of tries for a target. A useful shortcut: after n = 1/p tries, the chance is about 1 − e^(−1) ≈ 63%, not 100%.
Worked example
- Inputs: chance per try = 1% (p = 0.01), tries = 100.
- Chance of missing every time: 0.99¹⁰⁰ ≈ 0.36603.
- P(at least once) = 1 − 0.36603 ≈ 63.4%.
- Tries for 50%: ln 0.5 ÷ ln 0.99 ≈ 68.97, rounded up to 69 tries.
- Tries for 90%: ln 0.1 ÷ ln 0.99 ≈ 229.11, so 230; for 99%: ln 0.01 ÷ ln 0.99 ≈ 458.21, so 459. A fraction of a try is impossible, so the calculator always rounds up.
| Chance per try (%) | P(at least once) |
|---|---|
| 0.25 | 22.14% |
| 0.5 | 39.42% |
| 0.75 | 52.9% |
| 1 | 63.4% |
| 1.25 | 71.57% |
| 1.5 | 77.94% |
| 2 | 86.74% |
How to read the result
Doubling the tries does not double the chance: with 1% per try, 100 tries give 63% and 200 tries give about 87%. The last few percent are expensive: going from 90% to 99% takes twice as many tries as reaching 90%. Use the 90% or 99% figures when planning a budget, because the 50% figure means half of players will need more.
Common mistakes
- Multiplying p by n (1% × 100 = 100%), which overstates the chance.
- Believing you are "due" after many failures; with independent tries the next chance is still p.
- Ignoring pity or guarantee systems, which make later tries more likely to succeed.
Frequently asked questions
Is 1 − (1 − p)ⁿ the same as the binomial distribution?
Yes. It is 1 minus the binomial probability of zero successes in n trials.
What if the chance differs between tries?
Multiply each try's miss chance, (1 − p₁)(1 − p₂)…, and subtract the product from 1.
Why is 100 tries at 1% not a sure thing?
Each try can fail independently, and the chance of failing all 100 is still about 36.6%.
Sources: Ross, S. M., A First Course in Probability (Pearson), binomial random variables · Feller, W., An Introduction to Probability Theory and Its Applications, Vol. 1 (Wiley)