Probability Calculator
Calculate single event probabilities, joint probabilities (AND/OR), conditional probability, and odds ratios.
Probability Results
Frequently asked questions
How is single event probability calculated?
The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
What is the difference between independent and mutually exclusive events?
Independent events do not affect each other, so P(A and B) = P(A) x P(B) - two coin flips, for instance. Mutually exclusive events cannot both happen, so P(A and B) = 0 and P(A or B) = P(A) + P(B) - rolling a 3 and a 5 on one die. The two concepts are often confused; mutually exclusive events with non-zero probability are never independent.
How does conditional probability work?
P(A given B) = P(A and B) / P(B). It re-scales the probability of A to the world where B has already happened. Bayes' theorem rearranges this into P(A given B) = P(B given A) x P(A) / P(B), which is what lets you invert a conditional you know into one you want - the basis of most diagnostic-test reasoning.
What is the gambler's fallacy?
The belief that independent events somehow balance out - that after five reds, black is overdue. A fair roulette wheel has no memory, so the probability is unchanged every spin. The law of large numbers says the long-run proportion converges, not that short-run deviations get corrected.