Library / Statistics / STA-11
Statistics
Probability of at least one success
Chance that n independent attempts produce one or more successes.
- Ref
- STA-11
- Rev
- v1
- Field
- Statistics
- Result
- P
p is the chance of success on one attempt and n the number of independent attempts with the same p. The chance of failing every time is (1 − p)ⁿ, and the answer is its complement. For small p and moderate n the answer is close to np.
Chance that n independent attempts produce one or more successes. Many modest chances add up quickly.
Calculator
- Chance of no successes, (1 − p)ⁿ0.5905
- Attempts needed for a 90 % chance, rounded up22
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Five attempts at 10 % each give a 41 % chance of at least one success, not 50 %: chances do not simply add.
- Doubling the number of attempts does not double the chance; at 10 % you need 22 attempts to reach 90 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| p Probability of success each time | 0.001 to 0.99 | 0.00499 | 1 | 0.995 | +10% | +7.83 % |
| n Number of attempts | 1 to 100 | 0.1 | 1 | 0.9 | +1 | +14.4 % |
Sweep of Probability of success each time
| Probability of success each time | Probability |
|---|---|
| 0.001 | 0.00499 |
| 0.1246 | 0.486 |
| 0.2482 | 0.7599 |
| 0.3719 | 0.9022 |
| 0.4955 | 0.9673 |
| 0.6191 | 0.992 |
| 0.7428 | 0.9989 |
| 0.8664 | 1 |
| 0.99 | 1 |
Sweep one variable
Variables and how each one contributes
Probability
What the equation gives you. Everything else feeds it.
Probability of success each time
The chance of success on a single attempt. Raises the chance of at least one success, most strongly when the existing chance is still small.
Number of attempts
The number of independent attempts. Raises the chance, with diminishing returns: the first few attempts add most of the benefit.
Worked example with the values above
- Chance of failing once: 1 − p = 0.9.
- Chance of failing every time: 0.95 = 0.59049.
- P = 1 − 0.59049 = 0.40951.
Assumptions and limits
- Attempts are independent.
- The chance p is the same on every attempt.
Common mistakes
- Adding the probabilities (np) instead of using the complement.
- Using the formula when attempts affect each other.
Statistics and uncertainty
Assumes the attempts are independent. If events share a common cause, the real chance of at least one success is lower than this formula predicts.
Where it is used
- Cumulative chance of a detection or hit over repeated looks or shots.
- Chance that at least one of several redundant components works.
Maths you need
- Probability: the complement rule
- Independent events: multiplying probabilities