Calculate single event, complement, union (OR), and intersection (AND) probabilities for independent or dependent events.
Last updated: March 2026
Probability is a measure of how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). A probability of 0.5 means an event has a 50% chance of happening, like flipping a fair coin and getting heads.
This calculator supports four fundamental probability operations: single event (P(A)), complement (probability an event doesn't happen, 1 − P(A)), union (probability at least one of two events occurs, P(A ∪ B)), and intersection (probability both events occur simultaneously, P(A ∩ B)).
The calculator handles both independent events (where one event's outcome doesn't affect the other) and dependent events (where outcomes are related). For dependent events, you provide the conditional probability P(B|A), which is the probability of B occurring given that A has already occurred.
Drawing from a standard 52-card deck:
OR (union) means at least one event happens—you're looking for the probability of A, or B, or both. AND (intersection) means both events happen simultaneously. For example, rolling a 6 OR getting heads vs. rolling a 6 AND getting heads.
Uncheck independent when one event affects the other's probability. Classic example: drawing cards without replacement. If you draw an ace first (A), the probability of drawing another ace (B) changes because the deck composition changed.
P(B|A) is 'the probability of B given A'—a conditional probability. It's the probability of B occurring assuming A has already occurred. For example, if A = 'first card is an ace' and B = 'second card is an ace', then P(B|A) = 3/51 ≈ 0.0588.
When you add P(A) + P(B), you count the overlap (where both occur) twice. Subtracting P(A∩B) corrects this double-counting. Think of a Venn diagram: union is the total shaded area, so you add circles then subtract the overlap you counted twice.
No! Probabilities must be between 0 and 1 (or 0% to 100%). If you calculate a probability outside this range, there's an error—check that your events are properly defined and your inputs are correct.
The complement rule (P(A') = 1 − P(A)) is often easier for "at least one" problems. Instead of calculating all ways something can happen, calculate the probability it doesn't happen at all and subtract from 1. Example: P(at least one heads in 3 flips) = 1 − P(no heads).
For dependent events: P(B|A) = P(A∩B) / P(A). It's the joint probability divided by the probability of the condition. Intuitively: among all cases where A occurred, what fraction also have B?
No! Mutually exclusive events (can't both happen) are dependent. If A occurs, P(B) = 0. For independent events, P(B) stays the same regardless of A. Don't confuse these concepts—they're opposite in a way.
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