Calculator Inputs
Example Data Table
| Scenario | Trials | Successes | Probability | Exact P(X = k) | At Least P(X ≥ k) |
|---|---|---|---|---|---|
| Fair coin target | 10 | 5 | 0.50 | 0.2461 | 0.6230 |
| Lead conversion sample | 12 | 4 | 0.30 | 0.2311 | 0.5075 |
| Manufacturing pass rate | 20 | 8 | 0.40 | 0.1797 | 0.5841 |
| Rare defect detection | 15 | 2 | 0.10 | 0.2669 | 0.4510 |
| Email response check | 25 | 12 | 0.45 | 0.1564 | 0.4405 |
Formula Used
Binomial exact probability:
P(X = k) = C(n, k) × pk × (1 - p)n-k
Cumulative probability:
P(a ≤ X ≤ b) = Σ C(n, x) × px × (1 - p)n-x, summed from x = a to b
Expected value and spread:
Mean μ = n × p, Variance σ² = n × p × (1 - p), Standard deviation σ = √(n × p × (1 - p))
Probability score and z-score:
Probability Score = Selected Probability × 100. Z = (Observed Successes - Mean) / Standard Deviation
How to Use This Calculator
- Enter the total number of trials for the process you are studying.
- Provide the number of observed successes or the target count you want to evaluate.
- Enter the assumed success probability for one trial as a decimal between 0 and 1.
- Select the probability mode: exact, cumulative, tail, or range.
- If using range mode, enter the start and end success counts.
- Choose the number of decimal places for the output.
- Press the calculate button to display the result above the form.
- Review the score, interpretation, odds, confidence interval, and chart.
- Use the export buttons to download the results as CSV or PDF.
Frequently Asked Questions
1. What does the probability score mean?
It converts the selected probability into a percentage. A score of 72% means the chosen event definition has a 0.72 probability under the entered binomial assumptions.
2. When should I use exact mode?
Use exact mode when you need the probability of getting one specific success count, such as exactly 4 conversions in 12 outreach attempts.
3. What is the difference between at least and greater than?
At least includes the entered success count. Greater than excludes it. For example, at least 5 includes 5, while greater than 5 starts from 6.
4. Why does the calculator show a z-score?
The z-score shows how far the observed count is from the expected mean, measured in standard deviations. It helps identify whether the result is unusually low or high.
5. What is the Wilson interval used for?
It estimates a stable confidence interval for the observed success rate. Many analysts prefer it over a simple normal interval, especially with moderate sample sizes.
6. Can I use this for quality control or conversion analysis?
Yes. The model works well for repeated yes or no outcomes, including defect checks, lead conversions, pass rates, approval results, and response rates.
7. Why do I need a single-trial probability?
The binomial model assumes every trial shares the same success probability. That value is the base rate used to calculate exact, cumulative, and tail probabilities.
8. Does the graph change with larger trial counts?
Yes. The chart updates to reflect the entered distribution. For very large trial counts, points may be sampled to keep the page responsive and readable.