Calculator Input
This tool evaluates cumulative probability for X ≤ k in a binomial setting.
Example Data Table
| Trials (n) | Success Probability (p) | At Most Successes (k) | P(X ≤ k) | P(X = k) | Mean | Variance |
|---|---|---|---|---|---|---|
| 10 | 0.30 | 4 | 0.849732 | 0.200121 | 3.000000 | 2.100000 |
| 12 | 0.50 | 6 | 0.612793 | 0.225586 | 6.000000 | 3.000000 |
| 20 | 0.10 | 2 | 0.676927 | 0.285180 | 2.000000 | 1.800000 |
| 8 | 0.70 | 5 | 0.448226 | 0.254122 | 5.600000 | 1.680000 |
Formula Used
Binomial probability mass function:
P(X = x) = C(n, x) × px × (1 - p)n - x
At most probability:
P(X ≤ k) = Σ from x = 0 to k of P(X = x)
Combination term:
C(n, x) = n! / (x! × (n - x)!)
Summary statistics:
Mean = n × p, Variance = n × p × (1 - p), Standard deviation = √Variance
Use this model when each trial is independent, the success probability stays constant, and each trial has only two outcomes.
How to Use This Calculator
- Enter the total number of trials.
- Enter the probability of success for one trial.
- Enter the highest success count you want included.
- Choose how many decimal places should appear.
- Click calculate to view the cumulative result, graph, and table.
- Use CSV or PDF export for reporting or classroom notes.
Frequently Asked Questions
1. What does at most mean here?
At most means the random variable can be any value from zero through the chosen threshold. The calculator sums all included exact binomial probabilities to return one cumulative result.
2. When should I use a binomial model?
Use it when trials are independent, each trial has two outcomes, the number of trials is fixed, and the success probability stays the same every time.
3. What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) measures one exact outcome only. P(X ≤ k) adds that exact outcome plus every smaller outcome, creating a cumulative probability.
4. Why does the graph include both bars and a line?
The bars show exact probability for each success count. The line shows cumulative growth, helping you see where the at most threshold sits within the full distribution.
5. Can I enter a threshold larger than the trials?
Yes. If the threshold exceeds the number of trials, every possible outcome is included. The at most probability becomes 1 because nothing lies above the full support.
6. What do mean and variance tell me?
The mean gives the expected number of successes. Variance measures spread around that expectation. Together they help you judge how concentrated or dispersed the outcomes are.
7. Why are very large trials limited?
The calculator keeps the table and graph readable while protecting numerical stability. Extremely large trial counts can produce long tables and tiny probabilities that are harder to display clearly.
8. Who can benefit from this calculator?
Students, teachers, analysts, quality teams, and operations planners can use it for acceptance sampling, defect tracking, success forecasting, classroom examples, and probability reporting.