Calculator Form
Use this tool for binomial experiments where each trial has the same success probability and only two outcomes exist: success or failure.
Example Data Table
| Trials (n) | Exact Successes (k) | Success Probability (p) | Formula | Exact Probability |
|---|---|---|---|---|
| 10 | 4 | 0.30 | C(10,4) × 0.34 × 0.76 | 0.200121 |
| 12 | 5 | 0.40 | C(12,5) × 0.45 × 0.67 | 0.227030 |
| 8 | 2 | 0.25 | C(8,2) × 0.252 × 0.756 | 0.311462 |
Formula Used
Exact binomial probability:
P(X = k) = C(n, k) × pk × (1 - p)n-k
C(n, k) = n! / (k!(n-k)!)
- n = total number of trials
- k = exact number of successes
- p = probability of success on one trial
- 1 - p = probability of failure on one trial
This calculator assumes independent trials with a constant success probability. That setup matches the binomial model used in common experiments, surveys, defect counts, and repeated event analysis.
The extra outputs also report cumulative probabilities, expected successes, variance, standard deviation, and the most likely success count for the same distribution.
How to Use This Calculator
- Enter the total number of trials in the experiment.
- Enter the exact number of successes you want to evaluate.
- Type the success probability for one trial.
- Select decimal or percentage input format.
- Choose the number of decimal places for reporting.
- Press the calculate button to show the result above the form.
- Review the graph to compare your chosen outcome with the full distribution.
- Download CSV or PDF output for reporting or classroom use.
Frequently Asked Questions
1) What does exact probability mean here?
It means the probability of getting one specific number of successes, not fewer and not more. For example, exactly 4 successes in 10 trials is a single point on the full binomial distribution.
2) When should I use this calculator?
Use it when trials are independent, the success probability stays constant, and each trial has only two outcomes. Examples include coin flips, pass-fail tests, defect checks, and simple yes-no survey responses.
3) What is the difference between exact and cumulative probability?
Exact probability covers one precise success count. Cumulative probability adds several outcomes together, such as at most k successes or at least k successes. Both views are useful for interpretation.
4) Can I enter percentages instead of decimals?
Yes. Change the probability format to percentage and enter values like 30 for 30%. The calculator converts that value internally before applying the binomial formula.
5) Why does the chart highlight one bar?
The highlighted bar marks your selected exact success count. It helps you see how that single probability compares with other possible outcomes in the same distribution.
6) What does the expected successes value show?
Expected successes equals n × p. It is the long-run average number of successes you would anticipate across many repeated experiments with the same trial count and success probability.
7) Why can the most likely value differ from my selected exact count?
Your selected exact count is just one possible outcome. The most likely count, called the mode, is the outcome with the highest probability in the full binomial distribution.
8) Are CSV and PDF exports useful for teaching?
Yes. They make it easy to share worked examples, save reports, compare inputs, and document calculations for assignments, dashboards, and presentation material.