Probability Calculator

The online probability calculator computes P(A), the complement P(A'), the union P(A∪B), the intersection P(A∩B), and the conditional probability P(A|B) from the size of a sample space and event counts. It handles independent events, mutually exclusive events, and overlapping events, covering statistics homework, classroom demonstrations, and everyday risk analysis.

• P(A), P(A'), P(A∪B), P(A∩B), and P(A|B) are all computed from simple counts: the sample space size N and how many outcomes belong to each event.
• The relationship between two events is set as mutually exclusive, not mutually exclusive, or independent, to match how they actually interact.
• An overlap count resolves the intersection exactly when events can occur together.
• Results are formatted at a chosen decimal precision for reports, assignments, and presentations.
• Presets for coin flips, dice rolls, and a standard 52-card deck start from common probability models.

The probability calculator turns raw outcome counts into probabilities for single events and multiple events, using the size of the sample space and how many outcomes satisfy each event.

• N is the total number of possible outcomes in the scenario, the size of the sample space.
• k(A), k(B), and any further event counts state how many outcomes belong to each event.
• An advanced relationship setting treats two events as mutually exclusive, not mutually exclusive, or independent, to match how they actually interact.
• When events are not mutually exclusive, an overlap count k(A∩B) resolves the intersection exactly.
• Decimal precision is adjustable to format results for classwork, reports, or risk analysis.
• Presets for coins, dice, or cards start from a common probability model.

Counts should be whole numbers, the sum of mutually exclusive event counts should not exceed N, and an overlap is supplied only when events are not mutually exclusive.

Raw counts translate into probabilities that guide study, teaching, planning, and risk assessment.

• Verify homework by converting event counts into P(A), P(A'), P(A∪B), P(A∩B), and P(A|B).
• Teach statistics by demonstrating independent events, mutually exclusive events, complements, and conditional probability with results that update as counts change.
• Estimate business or project risk by modeling overlaps between triggers and outcomes to understand combined likelihood.
• Plan experiments by comparing theoretical probabilities to observed frequencies and refining sample sizes.
• Create study notes by copying formatted results and formulas for quick review before exams.

Probabilities are shown as fractions and decimals. P(A) is the chance that event A occurs and P(A') the chance that it does not. P(A∪B) is the chance that at least one event occurs, P(A∩B) the chance that both occur, and P(A|B) the chance that A occurs given that B already has. A relationship note states whether the events were treated as mutually exclusive, independent, or overlapping.

Cards: Modeling A as "draw a heart" and B as "draw a face card" from a standard deck gives N = 52, k(A) = 13, k(B) = 12. Treated as not mutually exclusive, with an overlap k(A∩B) = 3 for heart face cards, this gives P(A∪B) = 13/52 + 12/52 − 3/52 = 22/52 ≈ 0.423 and P(A|B) = (3/52) ÷ (12/52) = 0.25.

Quality control: Tracking A as "fails test 1" and B as "fails test 2" within a daily batch, with N as the batch size and k(A) and k(B) as the failure counts, plus the overlap between them, gives the chance an item fails both tests and the conditional probability of failing test 2 given it already failed test 1.

What is N? N is the total number of equally possible outcomes. Use the sample space size for coins, dice, cards, surveys, or batches.

When do I use independence? Independence applies when one event does not affect the probability of the other. If outcomes influence each other, treat them as dependent instead.

How do I handle overlaps? Events that are not mutually exclusive take an overlap count k(A∩B), which lets the union and the conditional probability compute correctly.

Why do results show fractions and decimals? Fractions show exact ratios, and decimals give readable values at a chosen precision. Both represent the same probability.