Two Events
Enter any two of the eight fields (P(A), P(B), P(A∩B), etc.). The tool infers the rest. Click regions on the Venn diagram to set the primary result.
Probability & odds
Probability Calculator
Multiple-event probability calculator: two-event solver (P(A∪B), P(A∩B), P(AΔB)), series of independent trials, and normal curve area. Venn diagram, Logic Trace, -inf/inf for tails.
By Jeff Beem
Updated
Probability Lab
Two Events: enter any two of the eight quantities to solve the rest. Series: independent trials. Normal: area between bounds (use -inf / inf for tails).
Fill any two valid fields (0–1); the rest are computed. Independence assumed when needed.
Formatting
Result
Enter at least two values (Two Events), or parameters for the selected mode.
Using the Probability Lab
Three modes cover different probability tasks. Two Events solves all eight quantities from any two inputs. Series handles independent trials. Normal Distribution finds area under the curve for any mean and standard deviation. Each mode has its own inputs and visual feedback.
Quick reference
Series
Use this mode when calculating probability across multiple events. Enter the number of events or trials (n) and the probability of success per event (p). The result is the probability that all succeed:
P(all success) = pn
Example: the probability of flipping heads 4 times in a row = 0.54 = 0.0625.Normal
Enter μ, σ, and bounds. Use -inf or inf for unbounded tails. The curve shades the region and shows raw values
x = μ + Zσ
plus Z-scores on the axis.Logic Trace
See the step-by-step math. Two Events shows the addition rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
and independence. Normal mode shows z-score conversion and Φ(z_R) − Φ(z_L)
CDF differences.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
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.- 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.
How the Math Works
The calculator's Two Events mode derives all eight probability quantities from any two valid inputs using the addition rule and complement identities. The core relationship is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). When only two inputs are given, the tool assumes independence, P(A ∩ B) = P(A) × P(B), to solve the system. Complements follow directly: P(A') = 1 − P(A). The symmetric difference is P(A Δ B) = P(A ∪ B) − P(A ∩ B). Series of Events mode computes the probability that all n independent trials succeed as p^n. Normal Distribution mode converts bounds to z-scores using z = (x − μ) / σ, then evaluates the area as Φ(z_R) − Φ(z_L) where Φ is the standard normal CDF. Worked example (Two Events): P(A) = 0.3, P(B) = 0.5. Assuming independence, P(A ∩ B) = 0.15, P(A ∪ B) = 0.65, P(A Δ B) = 0.50, and P((A ∪ B)') = 0.35.
How to Use This Calculator
Choose a mode from the tabs at the top. In Two Events mode, fill any two of the eight probability fields. P(A), P(B), P(A'), P(B'), P(A ∩ B), P(A ∪ B), P(A Δ B), or P((A ∪ B)'), and the calculator derives the remaining six. Click any result chip to highlight that region on the interactive Venn diagram. In Series of Events mode, enter the number of trials (n) and the probability of success per trial (p) to find P(all succeed) = p^n. In Normal Distribution mode, enter the mean (μ), standard deviation (σ), and left and right bounds; use -inf or inf for unbounded tails (for example, P(X > 1.5) uses left bound 1.5 and right bound inf). The shaded curve and Logic Trace show the z-score conversion and CDF computation step by step.
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.Probability of Multiple Events
When calculating probability across multiple events, the approach depends on what you're asking.
All events must occur (series/AND logic): Use the Series of Events mode. If each event has probability p and the events are independent, the probability that all n occur is pn. For example, the probability of rolling a 4 on a die three times in a row is (1/6)3 ≈ 0.0046.
At least one event occurs (union/OR logic): Use the Two Events mode with P(A ∪ B). For two independent events, P(A ∪ B) = P(A) + P(B) − P(A)×P(B). For example, if P(A) = 0.3 and P(B) = 0.4, then P(at least one occurs) = 0.3 + 0.4 − 0.12 = 0.58.
Neither event occurs (complement): P((A ∪ B)′) = 1 − P(A ∪ B). This is computed automatically in Two Events mode.
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
How do I calculate the probability of multiple events?
It depends on the relationship between the events. For independent events in series (e.g. "what's the probability all 5 succeed?"), use Series of Events mode: enter n (number of events) and p (probability per event) to get pn. For two events where you want the probability at least one occurs, use Two Events mode and read P(A ∪ B). For both events occurring, read P(A ∩ B). The calculator assumes independence when only two values are given, so no manual multiplication is needed, just enter any two known probabilities and the rest are derived automatically.
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.Mathematical Reference Note
Calculation Logic: This tool uses standard mathematical algorithms. While we strive for accuracy, errors in logic or user input can result in incorrect data.
Verification: Results should be cross-checked if used for important academic, professional, or personal calculations.
Standard Terms: This tool is provided free of charge and as-is. CalcRegistry provides no warranty regarding the accuracy or fitness of these results for your specific needs.
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