Probability Calculator

Probability quantifies how likely an event is to happen on a scale from 0 (impossible) to 1 (certain). To compute compound probabilities involving multiple events like A and B, foundational set theory rules apply. The intersection probability, or A and B, is calculated as P(A ∩ B) = A × B for independent events. Conversely, the union probability, or A or B, is found using the inclusion-exclusion principle: P(A ∪ B) = A + B - (A × B). For repeated trials over a custom number of times, exponents determine the cumulative likelihood of events happening always, never, or at least once.

Imagine you are managing an IT project where two independent subsystems have distinct failure probabilities. Subsystem A has a failure probability of 0.10 (10%), and Subsystem B has a failure probability of 0.05 (5%). First, let us find the probability that both fail simultaneously (A and B): multiply 0.10 by 0.05, resulting in 0.005 or 0.5%. Next, let us find the probability that at least one subsystem fails (A or B): add 0.10 and 0.05, then subtract their intersection (0.005), giving 0.145 or 14.5%. Finally, to find the chance that neither subsystem fails, compute (1 - 0.10) × (1 - 0.05), which equals 0.90 × 0.95 = 0.855 or 85.5%.

Always verify whether your events are independent or dependent before plugging values into standard formulas, as dependent events require conditional probability adjustments. Ensure your probabilities are always expressed as decimals or fractions between 0 and 1, rather than percentages, during intermediate mathematical steps. Double-check your inputs when evaluating repeated trials to avoid compounding errors over large sample sizes.