How this works
Probability quantifies how likely an event is, expressed as a number between 0 (impossible) and 1 (certain). For a single event A, the complement P(not A) = 1 − P(A). For two events, the rules depend on their relationship: if independent (one doesn't affect the other), P(A and B) = P(A) × P(B); if mutually exclusive (cannot both happen), P(A and B) = 0 and P(A or B) = P(A) + P(B). General case: P(A or B) = P(A) + P(B) − P(A and B).
Probability shows up everywhere: dice and card games, weather forecasts ("70% chance of rain"), insurance pricing, machine-learning model outputs, A/B test analysis, and risk management. Misunderstanding the relationship between events causes most real-world probability errors. The "gambler's fallacy" — believing past coin flips affect future ones — assumes dependence where none exists. The base-rate fallacy in medical testing assumes independence where there isn't. Always think carefully about whether your events are truly independent.
Odds are a related way to express the same information: odds = P / (1 − P). 50% probability is 1:1 odds; 80% is 4:1; 25% is 1:3. Bookmakers and statisticians use odds because likelihood ratios are mathematically cleaner than probabilities for some operations (especially Bayes' rule). The calculator displays both forms so you can use whichever fits your context.
The formula
P(A) and P(B) are probabilities between 0 and 1. Independence means P(A given B) = P(A) — knowing B happened doesn't change A's probability. Mutual exclusivity means A and B cannot both occur (e.g. rolling a 1 and a 6 on a single die at the same time).
Example calculation
- Two coin flips, both heads. Independent events with P = 0.5 each.
- P(both heads) = 0.5 × 0.5 = 0.25 (25%). P(at least one head) = 0.5 + 0.5 − 0.25 = 0.75.
Frequently asked questions
How do I tell if events are independent?
Independent means knowing one event happened gives no information about the other. Coin flips are independent because the coin has no memory. Drawing cards without replacement is NOT independent — once a card is gone, the deck is different. Real-world events are often partially correlated, which is why naive independence assumptions fail (e.g. assuming credit defaults are independent in mortgage portfolios — they're not, as 2008 showed).
What's the gambler's fallacy?
Believing past results affect future probabilities of independent events. "Black has come up 10 times in a row, so red is due!" — wrong, the wheel has no memory. Each spin is independent with the same 18/37 chance of red. The fallacy applies to roulette, dice, coin flips and lotteries. The opposite — "hot streak" thinking ("red has come up 5 times, it'll come up again!") — is equally wrong. Both fall apart because independent events don't self-correct.