2026-08-07 11 min

Probability Calculator

Calculate probabilities of two events, series of independent events, and normal distribution areas — with complements, unions, intersections, XOR, and a two-value solver.

Calculator

Probability of Two Events

P(A) Help

0 to 1

P(B) Help

0 to 1

Series of Independent Events

Series P(A)

Repeat Times A

Series P(B)

Repeat Times B

Normal Distribution

Mean (μ)

Standard Deviation (σ)

Left Bound (Lb)

Right Bound (Rb)

Results update automatically

Results update live as you type

Status:P(A∩B) 0.25P(A∪B) 0.75Normal P 0.818595

Probability Summary

P(A ∩ B)0.25

P(A ∪ B)0.75

P(A △ B) XOR0.5

P(Lb < X < Rb)0.818595

P(All Series)0.46656

Full Breakdown

Event	Formula	Probability
P(A ∩ B)	0.5 × 0.5	0.25
P(A ∪ B)	0.5 + 0.5 − 0.25	0.75
P(A △ B)	0.5 + 0.5 − 2×0.25	0.5
P(A')	1 − 0.5	0.5
P(B')	1 − 0.5	0.5
P((A∪B)')	1 − 0.75	0.25

Probability Solver for Two Events

Enter any 2 values to solve for the rest.

P(A)

P(B)

P(A')

P(B')

P(A∩B)

P(A∪B)

P(A△B)

Results

Normal

P(A ∩ B)

0.25

Both events occur: P(A) × P(B)

P(A ∪ B)

0.75

At least one occurs: P(A)+P(B)−P(A∩B)

P(A △ B) XOR

0.5

Exactly one occurs

P(A')

0.5

A does not occur: 1 − P(A)

P((A∪B)')

0.25

Neither event occurs

P(All Series Events)

0.46656

P(A)^2 × P(B)^3 for independent repeats

P(Lb < X < Rb)

0.818595

Normal(μ=0, σ=1) between bounds

Two-event results assume independence (product rule) with inputs clamped to 0–1. Normal areas difference the error-function CDF at standardized bounds; -inf/inf denote open tails.

Visualizations

Event Probabilities

0.3P(A∩B)

0.8P(A∪B)

0.5P(XOR)

0.5P(A')

0.5Series

0.8Normal

Outcome Split

A only25%

Both25%

B only25%

Neither25%

Union Coverage

P(A∪B)75%

Intersection vs Union

∩25%

Formula

Two-event results assume independence, so the intersection is the product and the union follows inclusion–exclusion. Series results chain that product across repeats. Normal areas standardize bounds to z-scores and difference the error-function CDF.

P(A∩B) = P(A)P(B) · P(A∪B) = P(A)+P(B)−P(A∩B) · P(A△B) = P(A)+P(B)−2P(A∩B)

Alternative Form

Series: P(all) = P(A)^a·P(B)^b · Normal: P = Φ((Rb−μ)/σ) − Φ((Lb−μ)/σ), Φ(z) = ½[1+erf(z/√2)]

Variables

Symbol	Name	Unit	Description
P(A), P(B)	Event probabilities	0–1	Likelihood of each independent event
∩	Intersection	probability	Both events occur
∪	Union	probability	At least one event occurs
△	Symmetric difference	probability	Exactly one event occurs (XOR)
μ, σ	Normal parameters	x-units	Center and spread of the Gaussian
z	Z-score	σ	(x−μ)/σ; standardizes any normal value
Φ	Normal CDF	probability	Left-tail area via the error function

Explanation

Independence makes combination a multiplication; dropping it requires conditional probabilities instead. The normal CDF has no closed form, so the error-function approximation carries the computation.

Derivation

Independence: P(A∩B) = P(A)P(B) by definition of independent trials. Union: count A plus B, subtract the double-counted overlap. XOR: union minus intersection. Normal density f(x) = (1/σ√2π)e^(−(x−μ)²/2σ²) integrates to Φ via erf; differencing Φ at two z-scores gives the between-bounds area.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Validate that P(A) and P(B) lie between 0 and 1.

0 ≤ P ≤ 1

P(A):0.5

P(B):0.5

Intermediate: probabilities are unit-interval values

Result: Inputs valid

Step 2

Find the intersection for independent events.

P(A∩B) = P(A) × P(B)

P(A):0.5

P(B):0.5

Intermediate: 0.5 × 0.5

Result: P(A∩B) = 0.25

Step 3

Apply inclusion–exclusion for the union.

P(A∪B) = P(A) + P(B) − P(A∩B)

P(A):0.5

P(B):0.5

P(A∩B):0.25

Intermediate: 0.5 + 0.5 − 0.25

Result: P(A∪B) = 0.75

Step 4

Multiply independent repeats for the series probability.

P(all) = P(A)^a × P(B)^b

P(A):0.8

Repeats:2

P(B):0.9

Intermediate: 0.8^2 × 0.9^3

Result: P(all) = 0.46656

Step 5

Standardize the normal bounds to z-scores.

z = (x − μ) / σ

μ:0

σ:1

Intermediate: zLb = (-2 − 0) / 1; zRb = (1 − 0) / 1

Result: Bounds mapped to the standard normal curve

Step 6

Subtract cumulative probabilities for the area between bounds.

P = Φ(zRb) − Φ(zLb)

μ:0

σ:1

Intermediate: CDF evaluated via the error function

Result: P = 0.818595

Final Answer

0.25

Worked Examples

See how the calculator works with different scenarios.

Scenario

Event A has probability 0.5 and event B has probability 0.5.

Inputs

P(A):0.5

P(B):0.5

Calculation

P(A∩B) = 0.25 P(A∪B) = 0.5+0.5−0.25 = 0.75 XOR = 0.5

Result

Intersection 0.25 · Union 0.75 · XOR 0.5

Interpretation

Three useful readings from two inputs: both happen a quarter of the time, at least one three-quarters.

Understanding Your Result

Your Result: P(A ∩ B): 0.25

Typical range: Intersections sit below each parent probability; unions sit above each; normal tail areas beyond ±2σ are under 5%.

What This Means

Each figure is a long-run frequency: 0.75 means the outcome occurs in about 75 of 100 identical trials.

Why It Matters

From insurance pricing to medical testing, decisions under uncertainty run on exactly these combinations.

Professional Interpretation

Confirm independence before trusting products; with dependence, replace P(A)P(B) with P(A)·P(B|A) throughout.

Recommendations

•Keep probabilities inside 0–1
•State the independence assumption explicitly
•Use complements for “at least one” shortcuts
•Prefer -inf/inf for one-sided normal questions
•Sanity-check union ≥ each parent probability
•Round reported probabilities to 4 decimals

Educational Content

Introduction

Probability quantifies uncertainty on a 0–1 scale. This calculator covers the three everyday needs: combining two events, chaining repeated trials, and measuring areas under the normal curve.

Concept Explanation

Independence factorizes joint likelihood into products; the normal curve standardizes any bell-shaped measurement to z-scores so one CDF answers every question.

Applications

▸Risk assessment for defaults and disasters

▸Reliability of multi-stage systems

▸Medical test interpretation via Bayes’ rule

▸Polling margins and election forecasts

▸Quality-control defect rates

▸Game and gambling odds analysis

Benefits

•Six two-event results from two inputs
•Series mode for repeated-trial chains
•Full normal-distribution area solver
•Two-value solver inverts the formulas
•Venn and bell-curve visual feedback

Limitations

•Two-event mode assumes independence
•Normal mode assumes Gaussian data
•No Bayes or conditional-probability mode
•erf approximation carries ~1e−7 error

Best Practices

•Verify independence before multiplying
•Draw the Venn regions for “exactly one” questions
•Standardize to z before consulting tables
•Use solver mode to audit published figures
•Report the assumed distribution with normal results

Common Misconceptions

Myth: P(A∪B) is just P(A)+P(B).

Truth: That double-counts the overlap; subtract P(A∩B) unless events are mutually exclusive.

Myth: A 95% interval has 95% chance of holding the truth.

Truth: It is a long-run capture rate over repeated sampling, not a bet on one interval.

Myth: Correlation equals causation.

Truth: Joint probability patterns need randomized trials before causal claims.

Myth: Rare events will not happen to me.

Truth: Small probabilities over many exposures become near-certainties — the law of large numbers.

Practical Tips

•Enter probabilities as decimals (0.5, not 50%)
•Use 1 − P for “does not happen” instantly
•Chain series repeats instead of manual powers
•Type -inf/inf for open normal tails
•Cross-check union against complements
•Try the solver with any two known values
•Round final answers, never intermediate steps

Common Mistakes

Adding probabilities for overlapping events

Subtract the intersection: P(A∪B) = P(A)+P(B)−P(A∩B).

Assuming independence without checking

With dependence use P(A∩B) = P(A)·P(B|A) instead of the product.

Entering percentages as whole numbers

This calculator takes 0–1 decimals; 50% is 0.5.

Confusing P(A∩B) with P(A|B)

Intersection is joint occurrence; conditional divides it by P(B).

Forgetting σ > 0 for normal areas

Standard deviation must be positive; check the input when areas vanish.

Assumptions

This calculator makes the following assumptions to provide estimates:

Independent Events: Two-event and series products assume independence.

Valid Probabilities: Inputs are clamped to the 0–1 interval.

Gaussian Data: Normal areas assume a true normal distribution.

Exact Repeats: Series trials share identical per-trial probabilities.

Numerical CDF: Normal areas use the erf approximation.

Limitations

Be aware of these limitations when using this calculator:

No Dependence Modeling: Correlated events need conditional inputs.

Two Events Only: Larger event sets need manual chaining.

Normal-Only Curves: Binomial, Poisson, and t distributions are out of scope.

Solver Coverage: The quick solver inverts the core pairs, not every combination.

Frequently Asked Questions

Answers to common questions about this calculator.

What is probability?

Probability measures the likelihood of an event as a number between 0 (impossible) and 1 (certain): favorable outcomes divided by total possible outcomes. It underpins statistics, risk assessment, and decision-making under uncertainty.

How do you calculate P(A∩B)?

For independent events, multiply: P(A∩B) = P(A) × P(B). For dependent events, use P(A) × P(B|A). The intersection can never exceed the smaller of P(A) or P(B).

How do you calculate P(A∪B)?

Use inclusion–exclusion: P(A∪B) = P(A) + P(B) − P(A∩B). For mutually exclusive events the intersection is zero, so it simplifies to P(A) + P(B).

What is the complement rule in probability?

P(A') = 1 − P(A). An event and its complement are mutually exclusive and exhaustive, so their probabilities always sum to 1.

What is the difference between independent and dependent events?

Independent events do not influence each other (coin flips); dependent ones do (drawing cards without replacement). Only independent events allow the simple product rule.

What is conditional probability?

P(A|B) = P(A∩B) / P(B): the probability of A given that B occurred. It is the engine behind Bayes’ Theorem, medical testing, and spam filters.

What is Bayes’ Theorem?

P(A|B) = P(B|A) × P(A) / P(B). It updates beliefs with new evidence and powers diagnostics, filtering, and machine-learning classifiers.

What is the normal distribution?

The symmetric bell-shaped Gaussian curve defined by mean μ (center) and standard deviation σ (spread). About 68% of data falls within ±1σ, 95% within ±2σ, 99.7% within ±3σ.

What is a z-score?

z = (x − μ) / σ: how many standard deviations a value sits from the mean. Positive means above average; negative means below. It standardizes any normal distribution to N(0,1).

What is the empirical rule (68-95-99.7 rule)?

For normal data, ≈68% lies within μ±1σ, ≈95% within μ±2σ, and ≈99.7% within μ±3σ. A quick check for spread and outliers.

What is the Central Limit Theorem?

Sample means become approximately normal as n grows (typically n > 30), whatever the population shape. It justifies confidence intervals and most hypothesis tests.

What is the difference between probability and odds?

Probability is favorable/total outcomes (0–1); odds compare occurrence to non-occurrence, P(A)/P(A’). P=0.75 equals odds of 3:1.