Probability of Three Events Calculator

How to Use the Three-Event Probability Calculator

Input the probability of each independent event as a decimal (0 to 1) or percentage (0% to 100%). The calculator instantly computes four key outcomes:

• Intersection (all three): Probability that events A, B, and C all happen simultaneously
• Union (at least one): Probability that one or more of the three events occur
• Exactly one: Probability that precisely one event happens while the others do not
• None: Probability that all three events fail to occur

The tool assumes events are independent—meaning the outcome of one does not influence the others. This applies to scenarios like rolling multiple dice, drawing cards with replacement, or assessing unrelated outcomes.

Probability Formulas for Three Independent Events

These formulas govern all calculations within the tool. They extend fundamental probability principles to handle three events simultaneously.

• P(A), P(B), P(C) — Individual probability of each event, ranging from 0 (impossible) to 1 (certain)
• P(A ∩ B ∩ C) — Joint probability of all three events occurring together
• P(A ∪ B ∪ C) — Probability of at least one event happening
• P(exactly one) — Probability that only one of the three events occurs
• P(none) — Probability that zero events occur

Understanding Independent Events and Probability Rules

Independence is the cornerstone assumption for this calculator. Two events are independent when the occurrence of one does not alter the likelihood of the other. Rolling a die twice, flipping a coin and drawing a card, or testing two unrelated systems are all independent events.

Key probability principles apply:

• Multiplication Rule: For independent events, joint probability equals the product of individual probabilities. This is why P(A ∩ B ∩ C) = P(A) × P(B) × P(C).
• Addition Rule: The union probability requires summing individual probabilities, then subtracting overlaps to avoid double-counting. With three events, this means subtracting pairwise intersections and adding back the triple intersection.
• Complement Rule: The probability an event does not occur is one minus its probability of occurring: P(A') = 1 − P(A).

Common Pitfalls When Computing Multi-Event Probabilities

Several mistakes commonly arise when calculating probabilities for multiple events.

1. Confusing intersection with union — The intersection (AND) requires all events to occur—probabilities multiply and result is typically small. The union (OR) allows any event to occur—probabilities combine via addition with overlap correction. Verify which scenario your problem requires before selecting a formula.
2. Assuming dependence when events are independent — If event B's probability changes based on whether A occurred, they are dependent, not independent. This calculator only handles independent events. Dependent events require conditional probabilities P(B|A), which this tool does not compute.
3. Forgetting to subtract intersections in union calculations — Simply adding three probabilities P(A) + P(B) + P(C) overcounts scenarios where two or more events overlap. Always apply the inclusion-exclusion principle: subtract pairwise intersections, then add back the triple intersection to correct for over-subtraction.
4. Treating decimal and percentage inputs interchangeably without conversion — If you input 25 intending the decimal 0.25 but the tool interprets it as 25%, your results will be drastically different (0.25 versus 0.0001). Verify your unit selection matches your input format.

Practical Examples of Three-Event Probability Scenarios

Example 1 – Quality Control: A factory produces items with a 95% pass rate for three independent quality checks. The probability all three checks pass is 0.95³ ≈ 0.8574 (85.74%). The probability at least one check fails is 1 − 0.8574 = 0.1426 (14.26%).

Example 2 – Medical Testing: Three independent diagnostic tests each have an 80% accuracy. The probability a patient is correctly identified by all three is 0.8³ = 0.512 (51.2%). The probability all three tests fail is 0.2³ = 0.008 (0.8%), making at least one success highly likely.

Example 3 – Lottery Odds: Winning three separate lotteries with individual odds of 1 in 1000 requires multiplying probabilities: (0.001)³ = 0.000000001 (one in one billion)—astronomically unlikely despite modest individual odds.