Calculator Inputs
Formula Used
For continuous random variables, conditional probability over intervals uses P(A|B) = P(A∩B) / P(B), provided that P(B) > 0.
Each interval probability is obtained from an integral of the density: P(L ≤ X ≤ U) = ∫LU f(x) dx. When the selected model has a closed-form cumulative distribution, the calculator evaluates this as F(U) − F(L).
The overlap region is the intersection of A and B. Therefore, P(A∩B) = ∫ over A∩B f(x) dx. The page also estimates E[X|B] and Var(X|B) numerically using Simpson integration on the conditioning interval.
How to Use This Calculator
- Select the probability model that matches your random variable.
- Enter the distribution parameters, such as mean and standard deviation.
- Define event interval A, the interval whose probability you want.
- Define conditioning interval B, the evidence or known range.
- Choose plot resolution and output precision.
- Press Calculate to show results above the form.
- Review P(A), P(B), P(A∩B), P(A|B), the graph, and conditional moments.
- Export the computed summary using CSV or PDF.
Example Data Table
Sample scenarios| Distribution | Parameters | Event A | Condition B | P(A) | P(B) | P(A|B) |
|---|---|---|---|---|---|---|
| Normal | μ = 0.00, σ = 1.00 | [-1.0000, 1.0000] | [-2.0000, 2.0000] | 0.6827 | 0.9545 | 0.7152 |
| Exponential | λ = 1.40 | [0.5000, 2.0000] | [0.0000, 3.0000] | 0.4358 | 0.9850 | 0.4424 |
| Uniform | min = 0.00, max = 10.00 | [3.0000, 7.0000] | [2.0000, 9.0000] | 0.4000 | 0.7000 | 0.5714 |
| Triangular | min = 0.00, mode = 4.00, max = 10.00 | [2.0000, 6.0000] | [1.0000, 8.0000] | 0.6333 | 0.9083 | 0.6972 |
FAQs
1) What does this calculator actually measure?
It computes probabilities for intervals under a continuous distribution, then applies conditioning. It shows P(A), P(B), P(A∩B), and the final conditional probability P(A|B).
2) What is the meaning of interval A?
Interval A is the event of interest. For example, it can represent acceptable output range, target tolerance, or a score band you want to evaluate.
3) What is the meaning of interval B?
Interval B is the known condition or evidence. It limits the sample space to values already known to have occurred, then recalculates the event probability inside that region.
4) Why can P(A|B) be undefined?
Conditional probability requires P(B) to be greater than zero. If the conditioning interval has zero probability under the selected model, division by zero would occur, so the conditional result is undefined.
5) Can I leave interval bounds empty?
Yes. A blank lower bound is treated as negative infinity, while a blank upper bound is treated as positive infinity. This is useful for one-sided probability questions.
6) Why are conditional mean and variance included?
They provide extra insight after conditioning. Once B is known, the distribution shifts, so the center and spread inside B can differ from the original model.
7) Does the graph show the conditional region?
Yes. The full density appears as the main curve. The conditioning region B is shaded, and the overlap A∩B is highlighted separately for quick visual interpretation.
8) When should I use each supported distribution?
Use Normal for symmetric continuous behavior, Exponential for waiting times, Uniform for equal-density ranges, and Triangular when only minimum, mode, and maximum are available.