Free Probability Calculator
P(A ∪ B), P(A ∩ B), and conditional probability from two events.
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Result
P(A ∪ B)
0.580000
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Short answer
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are independent, P(A ∩ B) = P(A) · P(B).
What is the Probability Calculator?
Basic probability arithmetic for two events, including independence and conditional probability.
How does the Probability Calculator work?
Union subtracts the intersection to avoid double-counting. Conditional divides the joint by the given event.
Formula
Variables
∪Union — 'or'∩Intersection — 'and'|Given (conditional)
Explanation
The inclusion-exclusion principle avoids double-counting the overlap.
Examples
Example 1: P(A) = 0.4, P(B) = 0.3, independent
P(A ∩ B) = 0.12; P(A ∪ B) = 0.58.
Applications
- Risk analysis
- Bayesian reasoning
- Card and dice games
Advantages
- Handles both independent and dependent cases
Limitations
- Two events only — extend manually for more
Common mistakes
- Adding probabilities when the events overlap (must subtract the intersection)
Tips
- P(A ∪ B) never exceeds 1, and never exceeds P(A) + P(B)
Related concepts
The Probability Calculator sits inside the Math Calculators hub, in the statistics & probability cluster. Describing data and quantifying uncertainty. Understanding the terms below makes the output easier to interpret and easier to compare against neighbouring measures.
Practical use cases and industry applications
Understanding Probability
Basic probability arithmetic for two events, including independence and conditional probability. Within math calculators, probability belongs to the statistics & probability cluster, where it shares terminology and assumptions with closely related tools.
Learning how probability is calculated
The inclusion-exclusion principle avoids double-counting the overlap. Working through the variables one at a time — ∪, ∩, | — makes the result reproducible by hand and easier to sanity-check.
Using probability to make a decision
Risk analysis Bayesian reasoning Card and dice games Because outputs depend on the assumptions you enter, run more than one scenario before committing to a figure.
How probability compares with related measures
distribution, dispersion, significance, sample all describe adjacent aspects of statistics & probability. Comparing this calculator's output against those measures — using the related tools listed on this page — prevents a single metric from being read in isolation.
Frequently asked questions
What if events are mutually exclusive?
Then P(A ∩ B) = 0, and P(A ∪ B) simplifies to P(A) + P(B).
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