Probability Calculator

Key Concepts

A few definitions tie the formulas together. The complement of A is the event 'A does not occur' and has probability 1 − P(A). The intersection A ∩ B is the event 'both A and B occur'; its probability equals P(A) × P(B) only when the events are independent. The union A ∪ B is 'A or B or both' and follows the inclusion-exclusion identity P(A) + P(B) − P(A ∩ B). The conditional P(A | B) = P(A ∩ B) / P(B) measures the probability of A inside the restricted sample space where B has already occurred. Two events are independent when knowing one tells you nothing about the other — equivalently, P(A | B) = P(A) and P(A ∩ B) = P(A) × P(B). Two events are mutually exclusive when they cannot both occur — P(A ∩ B) = 0 — so the union simplifies to P(A) + P(B). Independence and mutual exclusivity describe orthogonal relationships and (for non-trivial events) cannot both hold. Finally, Bayes' theorem inverts the conditioning direction: P(A | B) = P(B | A) × P(A) / P(B). It is the cornerstone of every test-interpretation, spam-filter, and updating-belief calculation.

Applications

• Insurance pricing — actuaries combine independent loss probabilities with the multiplication rule and union for portfolio-wide claim risk
• Medical test interpretation — Bayes' theorem inverts P(positive | disease) into P(disease | positive), preventing base-rate neglect
• Risk analysis and reliability engineering — series and parallel system reliability use P(A and B) and P(A or B) on component failure probabilities
• Weather forecasting — chance-of-rain aggregates conditional probabilities across atmospheric scenarios using the law of total probability
• Quality control — joint and conditional probabilities model defects across multiple production lines or inspection stages
• Gambling and game design — payout tables and house-edge calculations rely on accurate union and intersection probabilities for compound events
• Spam filtering and search ranking — naive Bayes classifiers use conditional independence assumptions to compute P(spam | features)

Common Mistakes

• Treating independent events as mutually exclusive (or vice versa). Independent means P(A | B) = P(A); mutually exclusive means P(A ∩ B) = 0. For events with positive probability, both cannot be true.
• Forgetting to subtract the intersection in the union formula. P(A or B) is NOT P(A) + P(B) unless the events are mutually exclusive — otherwise you double-count the overlap.
• Confusing P(A | B) with P(B | A). The two are equal only when P(A) = P(B). Use Bayes' theorem to convert: P(A | B) = P(B | A) × P(A) / P(B).
• Base-rate neglect — ignoring the prior P(A) when interpreting a conditional. A 99%-accurate test for a rare disease still produces mostly false positives because the prior P(disease) is so small.
• The gambler's fallacy — assuming past independent trials influence future ones. If P(heads) = 0.5 on a fair coin, the probability of heads on the next flip remains 0.5 regardless of how many tails just came up.
• Adding probabilities that exceed 1. If P(A) + P(B) > 1, the events cannot be mutually exclusive — there must be a non-zero intersection.