Library / Foundations: Maths / FMA-31
Foundations: Maths
Binomial probability
Chance of exactly k successes in n independent trials.
- Ref
- FMA-31
- Rev
- v1
- Field
- Foundations: Maths
- Result
- P(X=k) [%]
Use n independent trials with the same success probability p and two outcomes. Mean = np and variance = np(1 − p).
Count the ways to arrange k successes among n trials, and multiply by the chance of each arrangement.
Calculator
- Mean np1.5
- Standard deviation √(np(1−p))1.025
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.
- The most likely number of successes is close to np.
- Use it for repeated firings, test trials or component checks where each outcome is pass or fail.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (%) | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Number of successes | 0 to 10 | 0.243 | 36.01 | 35.77 | +10% | −7.11 pp |
| p Success probability | 0.05 to 0.5 | 2.143 | 34.55 | 32.41 | +undefined% | — |
| n Number of trials | 2 to 20 | 2.785 | 32.44 | 29.65 | +10% | +1.09 pp |
Sweep of Success probability
| Success probability | Probability (%) |
|---|---|
| 0.05 | 2.143 |
| 0.1063 | 8.059 |
| 0.1625 | 15.51 |
| 0.2188 | 22.82 |
| 0.275 | 28.82 |
| 0.3312 | 32.82 |
| 0.3875 | 34.5 |
| 0.4437 | 33.89 |
| 0.5 | 31.25 |
Sweep one variable
Variables and how each one contributes
Probability
What the equation gives you. Everything else feeds it.
Number of trials
How many trials. More trials spread probability across more outcomes.
Number of successes
The count you ask about. Probability peaks near np.
Success probability
Chance of success in each trial. Strong: probability peaks when p = k/n.
Worked example with the values above
- Ways to choose: C(5, 2) = 10.
- Chance of one arrangement: 0.32 × 0.73 = 0.03087.
- P = 30.87 %.
Assumptions and limits
- Independent trials with a constant p.
- Exactly two outcomes per trial.
Common mistakes
- Forgetting the C(n, k) factor.
- Using it when trials depend on each other.
Statistics and uncertainty
The standard deviation √(np(1−p)) tells how far results typically stray from np; observed frequencies converge on p as n grows, with error about √(p(1−p)/n).
Where it is used
- Probability of k hits in n shots.
- Number of failures in a batch of components.
Maths you need
- Combinations: n choose k
- Independent events: multiply probabilities