P(X = 6)
0.205078
20.508%
Probability
Binomial Distribution Calculator
Exact probabilities, tails, range checks, a distribution chart, CSV export, and binomial-test p-values in one worksheet.
P(X = k)P(X <= k)P(X >= k)RangeA/B test
About this calculator
Method, formulas, and limits.What this does
Computes exact binomial probabilities for a fixed number of trials and a success probability, including point, cumulative-tail, and inclusive-range values. It also provides a continuity-corrected normal approximation and an exact binomial test for A/B scenarios.
Who it is for
For students learning probability and statistics, analysts running A/B tests, and quality-control teams evaluating defect rates against acceptance thresholds.
How it works
Enter trials n, success probability p, and successes k. The calculator evaluates the probability mass function for every k and sums rows for cumulative and range queries. A/B mode treats observed successes as a binomial count under the hypothesized rate and returns exact p-values.
Limitations
The model assumes independent trials with a constant success probability. The normal approximation is generally reliable only when np and n(1-p) are both at least 5.
Key calculations
- Point probability
- P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
- Lower cumulative tail
- P(X <= k) = sum of P(X = i) for i from 0 to k
- Upper cumulative tail
- P(X >= k) = 1 - P(X <= k-1)
- Normal approximation
- z = (k ± 0.5 - np) / sqrt(np(1-p)), with continuity correction
Reference ranges
- Success probability p
- Common values include 0.5 for fair coins, 1/6 for fair dice, 0.02–0.05 for defect rates, and 0.01–0.10 for A/B conversion rates.
- Sample size n
- Small n suits classroom examples, moderate n suits quality control, and large n suits A/B tests and survey analysis.
- Normal approximation rule
- The approximation is generally safe when np >= 5 and n(1-p) >= 5. Values of p closer to 0.5 usually need a smaller n.
- P-value thresholds
- Common significance levels are 0.05, 0.01, and 0.10. Lower p-values indicate stronger evidence against the null hypothesis.
How to use it
- 1.Select a modeChoose Distribution for general probabilities or A/B test for hypothesis testing against a hypothesized rate.
- 2.Enter trial parametersEnter trials n, success probability p, and successes k. Presets provide quick starts for coins, dice, A/B tests, and quality control.
- 3.Choose a probability rangeUse the point and tail cards, or enable Range for an inclusive interval such as P(3 <= X <= 7).
- 4.Compare the normal approximationToggle continuity correction to compare the exact binomial result with the normal estimate.
- 5.Review the chartThe probability-mass chart shows the full distribution. Hover or tap a bar to inspect its probability.
- 6.Export the tableDownload the complete probability table as CSV for spreadsheet or statistical analysis.
It is the exact probability of getting exactly k successes in n independent trials when each trial has success probability p.
P(X = k) = C(n,k) × p^k × (1-p)^(n-k), where n is trials, k is successes, p is success probability, and C(n,k) is the binomial coefficient.
P(X <= k) sums outcomes from zero through k. P(X >= k) sums outcomes from k through n. They are used for lower- and upper-tail tests.
Yes. Enable range probability and enter the lower and upper success counts. The calculator sums every exact probability in that inclusive interval.
It is generally reliable when np >= 5 and n(1-p) >= 5. The calculator shows eligibility and lets you compare the approximation with the exact result.
Yes. It treats observed successes as a binomial count under a hypothesized p0 and computes exact one-sided or two-sided p-values.
It adjusts the discrete binomial distribution for the continuous normal curve by adding or subtracting 0.5 before calculating the z-score.
P(X < 6)
0.623047
62.305%
P(X <= 6)
0.828125
82.813%
P(X > 6)
0.171875
17.188%
P(X >= 6)
0.376953
37.695%
Mean
5
Variance
2.5
Std dev
1.5811
Skewness
0
Excess kurtosis
-0.2
Distribution table
| k | P(X=k) | P(X<=k) |
|---|---|---|
| 0 | 0.000977 | 0.000977 |
| 1 | 0.009766 | 0.010742 |
| 2 | 0.043945 | 0.054688 |
| 3 | 0.117188 | 0.171875 |
| 4 | 0.205078 | 0.376953 |
| 5 | 0.246094 | 0.623047 |
| 6 | 0.205078 | 0.828125 |
| 7 | 0.117188 | 0.945313 |
| 8 | 0.043945 | 0.989258 |
| 9 | 0.009766 | 0.999023 |
| 10 | 0.000977 | 1 |
Calculation steps
- 1.C(10,6) = 210
- 2.p^k = 0.5^6 = 0.015625
- 3.(1-p)^(n-k) = 0.5^4 = 0.0625
- 4.P(X=6) = C(10,6) x p^k x (1-p)^(n-k) = 0.205078125
Normal approximation
np = 5 and n(1-p) = 5
Exact
0.205078
Normal
0.204524
Error
0.27%
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