Answer first
What this calculator tells you
Calculate the chance of exactly k successes, and of k or fewer, in n independent trials. Check the odds of a run of successes before you treat a streak as skill or as bad luck. Formula: P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ; P(X ≤ k) = the sum of P(X = j) for j = 0 to k. At the worked-example inputs, the exactly k successes is 11.7%. Holding every other input steady, moving number of trials (n) from 8 to 12 moves the result from 5.4% to 21.9%.
Transparent method
The formula
Check the odds of a run of successes before you treat a streak as skill or as bad luck.
Worked example
Example inputs
How to interpret the result
Ten coin flips will land on exactly five heads only about a quarter of the time, and on exactly three about 12 percent. People expect the most likely outcome to be near-certain, and it never is. This page shows the chance of hitting an exact count and the chance of that count or fewer, which is the number you want when asking whether a result was unusually low.
At the worked-example inputs the exactly k successes is 11.7%. It rises with successes wanted (k) and falls as number of trials (n) and chance of success each trial increase.
These figures assume the model's conditions hold: independent trials with a fixed chance for the binomial, and a bell-shaped distribution for the normal. Real data that breaks those assumptions will not follow the calculated probability.
Before you rely on it
What to check
Check that the trials are independent and the chance is the same each time. Drawing cards without replacement breaks both rules.
The common error
Where people go wrong with binomial probability calculator
Reading the expected value as a promise. Five heads in ten flips is the average, yet the single most likely result still has less than a 25 percent chance of occurring.
Sensitivity evidence
How number of trials (n) changes the exactly k successes
Holding every other input at the worked-example value, moving number of trials (n) from 8 to 12 moves the exactly k successes from 5.4% to 21.9%: a spread of 16.5%, or 141% of the worked-example result.
| Number of trials (n) | Exactly k successes | k or fewer successes | Expected successes (n × p) |
|---|---|---|---|
| 8 | 21.9% | 36.3% | 4 |
| 9 | 16.4% | 25.4% | 4.5 |
| 10worked example | 11.7% | 17.2% | 5 |
| 11 | 8.1% | 11.3% | 5.5 |
| 12 | 5.4% | 7.3% | 6 |
Every input, tested
Which input moves the exactly k successes most
Of the 3 inputs, successes wanted (k) moves the exactly k successes most (16.1% across the range tested) and chance of success each trial moves it least (3.7%).
| Input | Tested from | To | Exactly k successes at each end | Swing |
|---|---|---|---|---|
| Successes wanted (k) | 2 | 4 | 4.4% to 20.5% | 16.1% (137%) |
| Number of trials (n) | 9 | 11 | 16.4% to 8.1% | 8.3% (71%) |
| Chance of success each trial | 48.0% | 52.0% | 13.6% to 9.9% | 3.7% (32%) |
Two variables at once
Exactly k successes by number of trials (n) and successes wanted (k)
Across the grid the exactly k successes runs from 0.293% to 24.6%. Moving number of trials (n) from 8 to 12 shifts it by 16.5% at the middle column, and moving successes wanted (k) from 1 to 5 shifts it by 23.6% at the middle row, so successes wanted (k) is the bigger lever here.
| Number of trials (n) \ Successes wanted (k) | 1 | 3 | 5 |
|---|---|---|---|
| 8 | 3.1% | 21.9% | 21.9% |
| 9 | 1.8% | 16.4% | 24.6% |
| 10 | 0.977% | 11.7% | 24.6% |
| 11 | 0.537% | 8.1% | 22.6% |
| 12 | 0.293% | 5.4% | 19.3% |
The highlighted cell is the worked example: 11.7%.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Number of trials (n) | 10 | How many independent attempts are made, up to 1,000. |
| Successes wanted (k) | 3 | The exact count of successes you are asking about. |
| Chance of success each trial | 50.0% | The probability that any single trial succeeds, from 0 to 100 percent. |
| Exactly k successes | 11.7% | |
| k or fewer successes | 17.2% | |
| Expected successes (n × p) | 5 | |
Inputs, definitions and assumptions
Number of trials (n)
How many independent attempts are made, up to 1,000. The prefilled worked-example value is 10.
Successes wanted (k)
The exact count of successes you are asking about. The prefilled worked-example value is 3.
Chance of success each trial
The probability that any single trial succeeds, from 0 to 100 percent. The prefilled worked-example value is 50.0%.
How to use this calculator
- 1Verify the inputs. Gather number of trials (n), successes wanted (k) and chance of success each trial from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the exactly k successes at 11.7%. Store your own version of it as Scenario A.
- 3Test one change. Start with successes wanted (k), the input with the biggest effect here: moving successes wanted (k) from 2 to 4 takes the exactly k successes from 4.4% to 20.5%, a swing of 137% of the worked-example figure.
- 4Check the extremes. At half the example successes wanted (k) (1.5) the exactly k successes is 4.4%; at double (6) it is 20.5%.
People also ask
Frequently asked questions
How do you calculate binomial probability?
P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ; P(X ≤ k) = the sum of P(X = j) for j = 0 to k. Enter chance of success each trial in percent (50 means 50%). At the worked-example inputs the exactly k successes is 11.7%.
What does the binomial probability result mean?
Check the odds of a run of successes before you treat a streak as skill or as bad luck. At the worked-example inputs the exactly k successes is 11.7%. It rises with successes wanted (k) and falls as number of trials (n) and chance of success each trial increase.
How much does number of trials (n) change the exactly k successes?
Holding every other input at the worked-example value, moving number of trials (n) from 8 to 12 moves the exactly k successes from 5.4% to 21.9%, a spread of 16.5%.
What are the limits of this binomial probability calculator?
These figures assume the model's conditions hold: independent trials with a fixed chance for the binomial, and a bell-shaped distribution for the normal. Real data that breaks those assumptions will not follow the calculated probability. The tables on this page test number of trials (n) only from 8 to 12; a value outside that range is not tabulated here.
Which input moves the exactly k successes most in the binomial probability calculator?
Ranked by how far each moves the exactly k successes across the range tested: successes wanted (k) (16.1%, 137%), number of trials (n) (8.3%, 71%) and chance of success each trial (3.7%, 32%).
If I double successes wanted (k) in the binomial probability calculator, does the exactly k successes double?
Doubling it from 3 to 6 takes the exactly k successes from 11.7% to 20.5%, which is 1.75 times the worked-example figure. So it grows, but by less than double. Halving it to 1.5 gives 4.4%.
How much does successes wanted (k) matter in the binomial probability calculator?
The worked example uses 3. With the other inputs left at the worked example, moving successes wanted (k) from 2 to 4 takes the exactly k successes from 4.4% to 20.5%, a swing of 137% of the worked-example figure.
How much does chance of success each trial matter in the binomial probability calculator?
The worked example uses 50.0%. With the other inputs left at the worked example, moving chance of success each trial from 48.0% to 52.0% takes the exactly k successes from 13.6% to 9.9%, a swing of 32% of the worked-example figure.
Which inputs change the k or fewer successes in the binomial probability calculator?
At the worked-example inputs it is 17.2%. Number of trials (n) takes it from 25.4% to 11.3%, successes wanted (k) takes it from 5.5% to 37.7% and chance of success each trial takes it from 20.7% to 14.1%.
Which inputs change the expected successes (n × p) in the binomial probability calculator?
At the worked-example inputs it is 5. Number of trials (n) takes it from 4.5 to 5.5 and chance of success each trial takes it from 4.8 to 5.2.
What is a normal distribution, and when does it apply?
A bell-shaped curve centered on the mean, with about 68 percent of values within one standard deviation and 95 percent within two. Heights and measurement errors are roughly normal. Incomes and waiting times usually are not.
What are independent trials?
Trials are independent when one outcome does not change the chance of the next. Coin flips are. Drawing cards without putting them back is not. The binomial model needs independence and a fixed chance of success.
Sources and evidence
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