Probability Calculator
50% probability of Event A
| Event | Favorable | Total | Probability |
|---|---|---|---|
| A | |||
| B |
How Probability Is Calculated
The probability of an event is defined as the number of favorable outcomes divided by the total number of equally likely possible outcomes: P(A) = favorable ÷ total, always a value between 0 and 1 (or 0% and 100%). This calculator also applies the standard rules for combining two events. For independent events, the multiplication rule gives P(A and B) = P(A) × P(B), and the addition rule gives P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events, the two events can never happen together, so P(A and B) = 0 and P(A or B) = P(A) + P(B).
Complements and Odds
The complement rule states that an event and its complement always sum to 1: P(A) + P(A′) = 1, which is why "probability it doesn't happen" is simply 1 minus the probability it does. Odds are a related but distinct way of expressing likelihood: odds in favor of A are favorable-to-unfavorable outcomes (P(A) : P(A′)), which is why a coin flip is "50% probability" but "1 to 1 odds," while a 25% probability corresponds to 1-to-3 odds, not 1-to-4.
A Common Misconception: Independent vs. Mutually Exclusive
These two terms are often confused but describe opposite situations. Independent events (like two separate coin flips) can absolutely happen together, and knowing one occurred tells you nothing about the other. Mutually exclusive events (like rolling a 2 or a 5 on one die) can never happen together by definition, which actually makes them dependent in a strict sense — if one happens, the other's probability drops to zero. Picking the right relationship above changes how P(A and B) and P(A or B) are computed. For counting the total number of ways outcomes can be arranged or chosen when working out favorable/total outcomes by hand, see the permutation and combination calculator. If your favorable/total inputs came from raw counts you need to simplify, the fraction calculator can help.
Frequently Asked Questions
What is the difference between independent and mutually exclusive events?
Independent events don't influence each other and can occur together (e.g., two separate coin flips), so P(A and B) = P(A) x P(B). Mutually exclusive events can never occur together (e.g., rolling a 2 or a 5 on one die), so P(A and B) = 0 and P(A or B) is simply P(A) + P(B).
How are odds different from probability?
Probability compares favorable outcomes to all possible outcomes (favorable / total), while odds compare favorable outcomes to unfavorable outcomes (favorable : unfavorable). A 50% probability equals 1-to-1 odds, but a 25% probability equals 1-to-3 odds, not 1-to-4, since only 1 of the 3 remaining outcomes needs to be excluded from the favorable count.