Binomial Distribution Calculator
Enter the number of independent trials, the chance of success in one trial and k, and get the chance of exactly k, fewer than k, at most k
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Estimate only. Not engineering or safety advice. Check the result against the applicable standard and the equipment data, and have a competent engineer review it before it is relied on. Terms of use.
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About this calculator
This page calculates P, exactly k, P, fewer than k, P, at most k and 5 more results from number of independent trials, probability of success in one trial and number of successes. Change any value or its unit and the result updates. The steps below show the working for the default values, so you can check each line against your own method.
Limits of this calculator
For a k-out-of-n system, count a unit that works as a success: the chance that at least k of n units work is the P(X ≥ k) result. For the number of ways to pick k items use the permutations and combinations calculator, and for two events use the probability calculator.
Cross-check: 34 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review.
Verification status: Engineering reference
Engineering reference. Formula and method documented and tested against separately coded reference calculations. Not yet Verified: the evidence listed on this page is still needed.
What this calculator checks
- The arithmetic of the published formula for the values you entered, in the units shown.
What remains unverified
- That the inputs suit your application. Take ratings, resistances and constants from the manufacturer data sheet or project specification.
Verification register entry
| Calculation author | Rahul Sharma (author) |
|---|---|
| Technical review | No independent technical review has been carried out |
| Individual professional approval | Not recorded |
| Formula reference | not recorded |
| Standard and edition | Not applicable: Pure mathematics or statistics; no electrical standard governs the method |
| Standard review state | Not applicable |
| Scope-screening note | not recorded |
| Cross-check test cases | 34 reference vectors (tests/reference_vectors.json) |
| Review date | not recorded |
| Next review date | not recorded |
| Status | Engineering reference |
Needed before this calculator can be marked Verified: reviewer name, qualification and review date; exact formula reference; formula source (textbook or standard); next review date.
Formula
Worked example
| Input | Value |
|---|---|
| Number of independent trials (n) | 20 |
| Probability of success in one trial (p) | 0.05 |
| Number of successes (k) | 2 |
- P(X = k) = nCk pᵏ (1 − p)ⁿ⁻ᵏ = 0.1886768013 (n = 20, p = 0.05, k = 2)
- P(X ≤ k) = 0.9245163262 P(X < k) = 0.7358395249
- P(X ≥ k) = 0.2641604751 P(X > k) = 0.0754836738
- Mean = n p = 1 Variance = n p (1 − p) = 0.95 Standard deviation = 0.9746794345
Assumptions
- Each of the n trials is independent, has only two outcomes, and has the same probability p of success
- Units that share a power supply, a cable or a cause of failure are not independent, so the result is optimistic for them
- The tail figures are added up term by term, not found as 1 minus the other side, so a very small tail keeps its precision
- Terms are calculated from sums of logarithms, accurate to about 12 significant figures for n up to 1000
- k must be a whole number from 0 to n
- The model has no wear-out, repair or ageing: p is the same for every trial
For a k-out-of-n system, count a unit that works as a success: the chance that at least k of n units work is the P(X ≥ k) result. For the number of ways to pick k items use the permutations and combinations calculator, and for two events use the probability calculator.
Inputs and results
| Input | Unit | Allowed or shown | Note |
|---|---|---|---|
| Number of independent trials (n) | (number) | 1 or more, up to 1,000 | A whole number from 1 to 1000, such as the number of lamps, units or samples. |
| Probability of success in one trial (p) | (number) | 0 or more, up to 1 | A number from 0 to 1. For 5% enter 0.05. Use the chance of the event you are counting, for example a lamp failing. |
| Number of successes (k) | (number) | 0 or more, up to 1,000 | A whole number from 0 to n. A success is the event with probability p. |
| Result | Unit | Allowed or shown | Note |
|---|---|---|---|
| P(X = k), exactly k | (number) | Shown to 10 decimal places | |
| P(X < k), fewer than k | (number) | Shown to 10 decimal places | |
| P(X ≤ k), at most k | (number) | Shown to 10 decimal places | |
| P(X > k), more than k | (number) | Shown to 10 decimal places | |
| P(X ≥ k), at least k | (number) | Shown to 10 decimal places | |
| Mean number of successes (n p) | (number) | Shown to 10 decimal places | |
| Variance (n p (1 − p)) | (number) | Shown to 10 decimal places | |
| Standard deviation | (number) | Shown to 10 decimal places |
How it is calculated
Each input is converted to the base unit of its quantity, the formula above is applied in the order shown in the worked example, and each result is shown in the unit you choose. Results are held at full precision and rounded only for display, except where a step selects a standard size, as the worked example shows. A value outside its allowed range is refused with a message instead of giving a result.
How the result changes with number of independent trials (n)
| Number of independent trials (n) | P(X = k), exactly k | P(X < k), fewer than k | P(X ≤ k), at most k |
|---|---|---|---|
| 2 | 0.0025 | 0.9975 | 1 |
| 5 | 0.021434375 | 0.9774075 | 0.998841875 |
| 10 | 0.0746347985 | 0.9138616441 | 0.9884964426 |
| 20 | 0.1886768013 | 0.7358395249 | 0.9245163262 |
| 40 | 0.2776716956 | 0.3990640651 | 0.6767357607 |
| 80 | 0.1445667121 | 0.0860537928 | 0.2306205049 |
| 200 | 0.0019322661 | 0.0004040281 | 0.0023362942 |
Other inputs are the same as in the worked example above.
Standards and further reading
Complex numbers, rectangular and polar form and angle conversion are standard mathematics, not the subject of a standard. Standards apply when the result feeds an electrical design, and they set the method there.
Frequently asked questions
What do I need to enter?
Enter number of independent trials (n), probability of success in one trial (p) and number of successes (k). Change the unit of a value with its selector and the result stays the same.
What does this calculator assume?
Each of the n trials is independent, has only two outcomes, and has the same probability p of success. Units that share a power supply, a cable or a cause of failure are not independent, so the result is optimistic for them. The tail figures are added up term by term, not found as 1 minus the other side, so a very small tail keeps its precision. Terms are calculated from sums of logarithms, accurate to about 12 significant figures for n up to 1000. k must be a whole number from 0 to n. The model has no wear-out, repair or ageing: p is the same for every trial.
Which calculators should I use next?
Common next steps are Permutations and Combinations Calculator (nPr, nCr), Probability Calculator (Two Events, A and B, and Repeated Trials), Standard Deviation and Variance and Percentage Calculator. The related links at the bottom of this page open them.
How accurate are the results?
Every figure comes from the same tested engine as the Elecmetric calculators and is meant for estimating. 34 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review. Check design values against the standard and the equipment data you work to.
Formula guides and reference
Longer explanations with worked examples and tables, written for the same topic.
- Probability, Combinations and Binomial Formulas: Probability formulas for reliability and test work: two events, repeated trials, permutations
- Mean and Standard Deviation Formulas: Statistics formulas for test readings: mean, median, mode, range, standard deviation, variance and standard error
- Measurement Uncertainty Formulas: Measurement uncertainty formulas: the standard uncertainty of a rectangular limit
More Engineering Mathematics calculatorsAll engineering calculatorsBinomial Distribution Calculator (Exactly k, At Most, At Least) in Elecmetric