Calculator Inputs
Example Data Table
| Distribution | Inputs | At most k | Result | Interpretation |
|---|---|---|---|---|
| Binomial | n = 10, p = 0.30 | 3 | 0.649611 | Chance of three or fewer successes in ten trials. |
| Poisson | λ = 4 | 2 | 0.238103 | Chance of at most two events in one interval. |
| Hypergeometric | N = 50, K = 12, n = 8 | 1 | 0.373169 | Chance of one or fewer marked items in eight draws. |
| Geometric | p = 0.22 | 4 | 0.629849 | Chance the first success occurs within four trials. |
| Discrete Uniform | min = 2, max = 12 | 7 | 0.545455 | Chance the result is seven or lower. |
Formula Used
At most probability: The calculator computes the inclusive cumulative probability P(X ≤ k).
- Binomial: P(X ≤ k) = Σ [C(n, x) · p^x · (1-p)^(n-x)], for x = 0 to k
- Poisson: P(X ≤ k) = Σ [e^-λ · λ^x / x!], for x = 0 to k
- Hypergeometric: P(X ≤ k) = Σ [C(K, x) · C(N-K, n-x) / C(N, n)] over valid x values
- Geometric: P(X ≤ k) = Σ [(1-p)^(x-1) · p], for x = 1 to k
- Discrete Uniform: P(X ≤ k) = favorable integers from a to k, divided by total integers from a to b
How to Use This Calculator
- Select the probability distribution that matches your problem.
- Enter the threshold value k for the phrase “at most.”
- Fill in the distribution parameters such as trials, rate, or bounds.
- Click Calculate Probability to generate the result above the form.
- Review the cumulative probability, exact probability at k, complement, and summary statistics.
- Use the Plotly graph to compare the probability mass and cumulative trend.
- Download the output as CSV for analysis or PDF for sharing.
FAQs
1. What does “at most” mean in probability?
It means the random variable is less than or equal to a chosen value. In notation, that is P(X ≤ k). The threshold is included, not excluded.
2. When should I use a binomial model?
Use binomial when you have a fixed number of independent trials, each trial has only success or failure, and the success probability stays constant.
3. When is the Poisson distribution more suitable?
Use Poisson when you model event counts in a fixed interval, especially when events occur independently and the average event rate is known.
4. Why does hypergeometric differ from binomial?
Hypergeometric handles sampling without replacement from a finite population. Because the composition changes after each draw, the success probability changes too.
5. What does the geometric option calculate?
It models the trial number of the first success. An at-most result shows the chance that the first success happens on or before trial k.
6. Why show both P(X ≤ k) and P(X = k)?
The cumulative value answers the at-most question. The exact value isolates the single threshold point, which helps compare marginal and cumulative behavior.
7. What does the complement probability represent?
It represents P(X > k), the chance of exceeding the threshold. This is useful for risk limits, service targets, and tail-event checks.
8. Can I export the result for reporting?
Yes. The calculator includes CSV export for structured data and PDF download for a quick report snapshot with your computed summary.