🎯 Binomial Probability Calculator
Calculate exact, cumulative, or upper-tail binomial probabilities for independent yes/no trials.
📊 Statistics📐 Formula included🧠 Practice questions✓ Free
Please enter valid values.
Formula and Explanation
Calculate exact, cumulative, or upper-tail binomial probabilities for independent yes/no trials.
| Concept | Formula or rule |
|---|---|
| Binomial Probability Calculator | P(X=k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ |
How to Use This Calculator
- Enter the sample, counts, probabilities, or summary statistics requested above.
- Keep paired lists in the same order and separate values with commas.
- Select the correct test direction or sample/population option when available.
- Press Calculate and interpret the result together with the supporting statistics.
Worked Example
Example: For 10 fair coin flips, the chance of exactly 5 heads is C(10,5)×0.5¹⁰.
✓ P(X=5) ≈ 0.2461
Where This Is Used
School & coursework
Check homework, understand formulas, and verify step-by-step statistics exercises.
Research
Explore estimates, uncertainty, distributions, and relationships before using full statistical software.
Business & data analysis
Summarize variation, compare groups, and communicate data-driven findings clearly.
Common Mistakes to Avoid
⚠️
Using changing probabilities
Binomial trials require the same success probability.
Binomial trials require the same success probability.
⚠️
Counting dependent trials as independent
Dependence changes the distribution.
Dependence changes the distribution.
⚠️
Entering percentages as whole numbers
Use 0.4 for 40%, not 40.
Use 0.4 for 40%, not 40.
Practice Questions
1. For n=1 and p=0.3, what is P(X=1)?
Answer: 0.3
2. What is the mean for n=20, p=0.5?
Answer: 10
3. Can k exceed n?
Answer: No.
Frequently Asked Questions
What conditions define a binomial model?▾
Fixed n, two outcomes, constant p, and independent trials.
What is the mean?▾
np.
What is the variance?▾
np(1−p).