Calculator Inputs
Choose a distribution, select a probability mode, enter the required parameters, and calculate exact, cumulative, tail, or interval probabilities for X.
Example Data Table
| Example | Distribution | Inputs | Question | Interpretation |
|---|---|---|---|---|
| Quality checks | Binomial | n = 10, p = 0.50 | P(X = 4) | Chance of exactly four successes in ten trials. |
| Call arrivals | Poisson | λ = 3.5 | P(X ≤ 2) | Chance of observing at most two arrivals. |
| First sale timing | Geometric | p = 0.30 | P(X ≥ 5) | Chance the first success takes five or more trials. |
| Sample inspection | Hypergeometric | N = 50, K = 15, n = 10 | P(2 ≤ X ≤ 4) | Chance the sample contains between two and four successes. |
| Test scores | Normal | μ = 50, σ = 8 | P(45 ≤ X ≤ 60) | Chance a score falls inside the selected range. |
Formula Used
Binomial
P(X = x) = C(n,x) px(1-p)n-x
Use binomial when you have a fixed number of independent trials and each trial has the same success probability.
Poisson
P(X = x) = e-λ λx / x!
Use Poisson for counts occurring over a time, distance, area, or volume when the average rate is known.
Geometric
P(X = x) = p(1-p)x-1
Use geometric when X is the trial number of the first success in repeated independent trials.
Hypergeometric
P(X = x) = [C(K,x) C(N-K,n-x)] / C(N,n)
Use hypergeometric for sampling without replacement from a finite population containing known successes.
Normal
f(x) = (1 / σ√(2π)) e-0.5((x-μ)/σ)²
Cumulative and interval results use the normal CDF. Exact normal results are approximated with continuity correction around the chosen x value.
How to Use This Calculator
- Select the probability distribution that matches your problem setup.
- Choose the probability type: exact, at most, at least, or between.
- Enter the distribution parameters and your x value or range.
- Press the calculate button to show the result above the form.
- Review the probability, percentage, mean, variance, and chart highlight.
- Use the CSV or PDF buttons to save the result summary.
FAQs
1) What does X mean in this calculator?
X is the random variable being measured. It could represent successes, arrivals, the trial of first success, sampled defects, or a continuous score.
2) Which distributions are included?
The calculator supports binomial, Poisson, geometric, hypergeometric, and normal models. That covers many common discrete and continuous probability tasks.
3) When should I use exact probability?
Use exact mode when you need one specific value, such as exactly four defects, exactly two arrivals, or exactly one score band near a target.
4) Why do some distributions need whole-number x values?
Discrete models count events or successes, so valid x values are integers. For cumulative or tail modes, the calculator floors or ceils boundaries when appropriate.
5) How is exact probability handled for the normal distribution?
A continuous normal variable has zero probability at one exact point. The calculator uses continuity correction around x to provide a practical approximation.
6) What does the highlighted graph show?
The graph marks the exact bars or shaded curve region that belongs to your selected probability. It helps you see the event rather than only reading the number.
7) Can I export the result?
Yes. Use the CSV button for spreadsheet-friendly output and the PDF button for a clean report summary that includes the displayed metrics.
8) How can I verify the result manually?
Check the chosen formula, confirm the parameter meanings, and compare the displayed mean, variance, and support. For simple cases, you can also sum exact probabilities directly.