Probability Calculator: Union, Intersection & Normal Distribution
Free probability calculator: two-event solver with 8-field inference, Venn diagram, normal curve area, and series of independent trials. P(A∪B), P(A∩B), P(AΔB). Use -inf and inf for tails.
What This Calculator Does and Who It's For
- Who it's forStudents learning set theory and probability; anyone computing union (OR), intersection (AND), or symmetric difference; users needing normal curve area or tail probabilities. Ideal for "probability of A or B," "P(A and B)," and "area under normal curve" queries.
- Trust and scopeAll calculations run in your browser. Standard normal CDF uses the error function. Two-event solver assumes independence when only two values are given. Formatting options (sig figs, scientific notation) handle very small probabilities.
This probability calculator (Probability Lab) supports three modes. Two Events: Enter any two of P(A), P(B), P(A′), P(B′), P(A ∩ B), P(A ∪ B), P(A Δ B), and P((A ∪ B)′)—the tool infers the remaining six using the addition rule and assumes independence when needed. Series of Events: Compute
P(all success) = pn
for n independent trials. Normal Distribution: Find the area between bounds for a given mean and standard deviation; use -inf or inf for tail probabilities. An interactive Venn diagram highlights the selected region when you click a result.Union, Intersection, and the Addition Rule
P(A ∪ B) (union) is the probability that at least one of A or B occurs. P(A ∩ B) (intersection) is the probability that both occur. The addition rule:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
For independent events: P(A ∩ B) = P(A) × P(B)
so P(A ∪ B) = P(A) + P(B) − P(A)×P(B)
The symmetric difference: P(A Δ B) = P(A ∪ B) − P(A ∩ B)
gives the probability of A or B but not both. This calculator derives all eight quantities from any two valid inputs.Normal Distribution: Area Between Bounds
For a normal distribution with mean μ and standard deviation σ, the probability that X falls between L_b and R_b is
P(L_b ≤ X ≤ R_b) = Φ(z_R) − Φ(z_L)
where z = (x − μ) / σ
and Φ is the standard normal CDF. Enter -inf for the left bound to get a right tail (e.g. P(X > 2)); enter inf for the right bound to get a left tail. The curve shades the region and displays dynamic X-axis labels from μ and σ.Venn Diagram and Click-to-Highlight
The Venn diagram shows two circles (A and B). Click any result chip—P(A), P(B), P(A ∩ B), P(A ∪ B), P(A Δ B), or P((A ∪ B)′)—to highlight that region. The primary result card updates to show the selected probability. This helps you see exactly which area corresponds to union, intersection, symmetric difference, or the complement of the union.
Probability Calculator FAQ
? What is P(A ∪ B) vs P(A ∩ B)?
P(A ∪ B) is the probability that at least one of A or B occurs (union). P(A ∩ B) is the probability that both A and B occur (intersection). The addition rule:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
For independent events: P(A ∩ B) = P(A) × P(B)
This Probability Lab lets you enter any two of the eight quantities to solve the rest, assuming independence when needed.? How does the Two-Event Solver infer missing values?
When you fill any two valid fields among P(A), P(B), P(A′), P(B′), P(A ∩ B), P(A ∪ B), P(A Δ B), and P((A ∪ B)′), the calculator derives the remaining six using the addition rule and complement identities. It assumes independence when needed:
P(A ∩ B) = P(A) × P(B)
Click a result chip to highlight its region on the Venn diagram and set it as the primary result.? What is P(A Δ B) (symmetric difference)?
P(A Δ B) is the probability of outcomes in A or B but not both—the "exclusive or" of the two events. Algebraically:
P(A Δ B) = P(A ∪ B) − P(A ∩ B)
or equivalently P(A Δ B) = P(A) + P(B) − 2×P(A ∩ B)
In the Venn diagram, it corresponds to the two "moons" outside the overlap. The Probability Lab highlights this region when you click the P(A Δ B) chip.? How do I find area under the normal curve?
In Normal Distribution mode, enter the mean (μ), standard deviation (σ), and left/right bounds (L_b, R_b). The tool converts bounds to z-scores:
z = (x − μ) / σ
then computes the area as Φ(z_R) − Φ(z_L)
using the standard normal CDF. Use -inf or inf for tail probabilities (e.g. P(X > 1.5) = area from 1.5 to infinity). The curve shades the region and shows dynamic X-axis labels from μ and σ.? When should I use Series of Events mode?
Use Series of Events when you have n independent trials and want the probability that all succeed. Formula:
P(all success) = pn
where p is the probability of success per trial. Example: probability of 5 heads in a row with a fair coin = 0.5^5 = 0.03125. This assumes independence between trials.