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 given number of trials and success probability, including point probabilities, cumulative lower-tail and upper-tail values, and inclusive range probabilities. It also provides a continuity-corrected normal approximation and an exact binomial test for A/B testing scenarios.
Who it is for
Students learning probability and statistics, data analysts running A/B tests, and quality control teams evaluating defect rates against acceptance thresholds.
How it works
Enter the number of trials n, success probability p, and the target number of successes k. The calculator evaluates the binomial probability mass function for each k and sums individual probabilities for cumulative and range queries. For the A/B test mode, it treats observed successes as a binomial count under the hypothesized rate and returns exact one-sided and two-sided p-values.
Limitations
Assumes independent trials with a constant success probability, which may not hold in real-world sampling without replacement. The normal approximation is only reliable when np and n(1-p) are both at least 5.
Key calculations
- Point Probability
- The exact probability of exactly k successes. P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
- Cumulative Lower-Tail
- The probability of k or fewer successes. P(X ≤ k) = Σ P(X = i) for i = 0 to k
- Cumulative Upper-Tail
- The probability of k or more successes. P(X ≥ k) = 1 - P(X ≤ k-1)
- Normal Approximation
- Approximates binomial probabilities using the normal distribution with continuity correction. z = (k ± 0.5 - np) / sqrt(np(1-p))
Reference ranges
- Success Probability p
- Common values: 0.5 for fair coin flips, 1/6 ≈ 0.167 for fair dice, 0.02–0.05 for quality control defect rates, and 0.01–0.10 for typical A/B test conversion rates.
- Sample Size n
- Small n (10–30) for classroom examples, moderate n (50–200) for quality control, and large n (500–5000) for A/B tests and survey analysis.
- Normal Approximation Rule
- The approximation is generally safe when np ≥ 5 and n(1-p) ≥ 5. The closer p is to 0.5, the smaller n can be for a reliable approximation.
- P-Value Thresholds
- Common significance levels: 0.05 for standard hypothesis tests, 0.01 for stricter tests, and 0.10 for exploratory analysis. Lower p-values indicate stronger evidence against the null hypothesis.
How to use it
- 1.Select calculator modeChoose Basic for general binomial probabilities or A/B Test for hypothesis testing against a hypothesized rate.
- 2.Enter trial parametersType the number of trials n, success probability p, and the target success count k. Use presets like Coin, Dice, or Quality Control for quick starts.
- 3.Choose probability typeSelect P(X = k) for an exact point, P(X ≤ k) for cumulative, P(X ≥ k) for tail, or enable Range for an inclusive interval like P(3 ≤ X ≤ 7).
- 4.Toggle normal approximationTurn on the normal approximation to compare exact binomial results with the continuity-corrected normal estimate side by side.
- 5.Review the distribution chartThe probability mass chart visualizes the full distribution. Hover or tap bars to see individual probabilities.
- 6.Export the distribution tableDownload the full probability table as CSV for further analysis in spreadsheet or statistical software.
It is the exact probability of getting exactly k successes in n independent trials when each trial has a constant success probability p. For example, the probability of getting exactly 6 heads in 10 fair coin flips would be P(X = 6) with n=10 and p=0.5.
The formula is P(X = k) = C(n,k) × p^k × (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the success probability per trial, and C(n,k) is the binomial coefficient (n choose k) counting how many ways k successes can occur in n trials.
P(X ≤ k) is the lower-tail cumulative probability of k or fewer successes, which sums P(X=0) through P(X=k). P(X ≥ k) is the upper-tail cumulative probability of k or more successes, which sums P(X=k) through P(X=n). These are used for one-sided hypothesis tests and confidence intervals.
Yes. Enable range mode and enter the lower and upper success counts. The calculator sums each exact binomial probability for every integer in that inclusive range, giving you the total probability that the number of successes falls between your specified bounds.
The normal approximation to the binomial is generally reliable when np ≥ 5 and n(1-p) ≥ 5. The calculator shows you whether this condition is met, lets you toggle the continuity correction (adding or subtracting 0.5 before calculating the z-score), and displays the approximation alongside the exact value so you can compare them directly.
Yes. The A/B test mode treats the observed number of successes as a binomial count under a hypothesized success rate p0. It computes exact one-sided (greater-than or less-than) and two-sided p-values, which tell you how likely the observed result would be if the true rate were actually p0.
The continuity correction adjusts the discrete binomial distribution to better match the continuous normal distribution by adding or subtracting 0.5 from k before computing the z-score. The formula becomes z = (k ± 0.5 - np) / sqrt(np(1-p)), which typically improves the approximation accuracy, especially for small n or extreme p.
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 chart
11 outcomes
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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