Probability Tool
Explore binomial, normal, Poisson, hypergeometric, and geometric probabilities alongside event relationships and combinatorial counts. Choose exact, cumulative, tail, or interval queries, inspect expected values and charts, review data tables, and export results.
Table of Contents
Highlights
- Seven modes covering distributions, event probabilities, and combinatorics
- Exact, cumulative, tail, and interval probability queries
- Interactive distribution charts with highlighted probability regions
- Expected values, variability statistics, and detailed data tables
- Export results as CSV or JSON and charts as SVG or PNG
How Probability Tool Works?
Probability quantifies the chance of an event, with values between 0 and 1. Select a probability model and enter its parameters; the result is calculated for an exact event, a cumulative threshold, a tail, or an interval. Discrete models use probability mass functions (PMFs) and cumulative sums. The normal model uses a continuous cumulative distribution function (CDF).
| Model | Inputs | Key mathematical relationship |
|---|---|---|
| Binomial | Trials n, success chance p, successes k | P(X = k) = C(n,k)pk(1 − p)n−k |
| Poisson | Average rate λ, count k | P(X = k) = e−λλk/k! |
| Hypergeometric | Population N, successes K, draws n, observed successes k | P(X = k) = C(K,k)C(N−K,n−k)/C(N,n) |
| Geometric | Success chance p, first successful trial k | P(X = k) = (1 − p)k−1p |
| Normal | Mean μ, standard deviation σ, boundary x or interval [x,b] | Probabilities are areas under the normal density curve |
The binomial model, for example, calculates the probability of exactly k successes across n independent trials:
Here the binomial coefficient represents the number of possible success arrangements: C(n,k) = n!/[k!(n−k)!]. Cumulative questions sum individual probabilities. For instance, P(X ≥ k) = 1 − P(X ≤ k−1), while inclusive discrete intervals use P(a ≤ X ≤ b) = F(b) − F(a−1).
For normally distributed measurements, standardise a boundary using the z-score, then evaluate the normal CDF:
Combinatorics supports combinations and permutations with or without repetition, plus factorials. Event operations use P(A), P(B), and P(A ∩ B), for example P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The independence option displays the implied overlap P(A)P(B); it does not test independence against observed data. Outputs include probabilities or arrangement counts, expected values where relevant, variance or supplementary statistics, plotted distributions, and tabulated values.
Practical Use Cases
Quality control analysts: Estimate the chance of finding a particular number of defective items across independent inspections using the binomial model, or apply the hypergeometric model when sampling a finite batch without replacement.
Customer support managers: Model the probability of receiving a specified number of calls within a period using a Poisson average arrival rate, provided arrivals are reasonably independent and the rate is stable.
Teachers and students: Explore normal-tail probabilities, expected values, cumulative distributions, and the visual difference between PMFs and densities while studying introductory statistics.
Researchers: Calculate the chance of a first success occurring on a particular trial with the geometric distribution, or examine unions, intersections, conditional probabilities, and mutually exclusive event regions.
Event planners and software testers: Count possible teams, selections, ordered arrangements, or repeated choices using combinations, permutations, and factorials.
Accuracy and Limitations
Results use JavaScript floating-point arithmetic, logarithmic factorial calculations, cumulative sums, and an approximation to the error function for normal probabilities, so rounding and extreme-tail errors are possible. Large combinatorial counts can overflow or lose integer precision; displayed probabilities are approximate rather than exact symbolic fractions. Some cumulative calculations stop after a fixed iteration cap for very large inputs, and charts show only a selected or sampled range rather than every possible outcome. The models assume their stated conditions, such as independent binomial trials, constant-rate Poisson events, or hypergeometric sampling without replacement. Normal probabilities at a single point are zero, and the independence option reports an implied overlap rather than a statistical independence test.
