Calculate probability for single events, multiple events (AND / OR), conditional probability, combinations, and permutations.
Enter the number of favorable outcomes and total outcomes to calculate the probability.
Enter the probability of each event (as a decimal or fraction) and choose whether they are independent.
Calculate P(A|B) ā the probability of A given that B has already occurred.
Calculate nCr ā the number of ways to choose r items from n items where order does not matter.
Calculate nPr ā the number of ways to arrange r items from n items where order matters.
Probability is the measure of how likely an event is to occur, expressed as a number between 0 and 1 (or 0% to 100%). A probability of 0 means the event is impossible, while a probability of 1 means it is certain. Probability is used in statistics, finance, science, games, and everyday decision-making.
| Type | Formula | Example |
|---|---|---|
| Single Event | P(A) = favorable Ć· total | P(heads) = 1/2 = 0.5 |
| AND (independent) | P(Aā©B) = P(A) Ć P(B) | P(heads AND heads) = 0.5 Ć 0.5 = 0.25 |
| OR (independent) | P(AāŖB) = P(A) + P(B) ā P(Aā©B) | P(A or B) where A=0.3, B=0.4 |
| OR (mutually exclusive) | P(AāŖB) = P(A) + P(B) | P(1 or 2 on a die) = 1/6 + 1/6 |
| Conditional | P(A|B) = P(Aā©B) Ć· P(B) | P(ace | red card) = 0.02 Ć· 0.5 = 0.04 |
A combination counts selections where order does not matter (choosing a team). A permutation counts arrangements where order does matter (ranking contestants). For example, choosing 3 from 5: nCr = 10 ways, but nPr = 60 arrangements.
Two events are mutually exclusive if they cannot both happen at the same time. For example, rolling a 1 and rolling a 2 on a single die ā you can't get both at once. For mutually exclusive events, P(A OR B) = P(A) + P(B).
Two events are independent if the outcome of one does not affect the other. For example, flipping a coin twice ā the first flip doesn't affect the second. For independent events, P(A AND B) = P(A) Ć P(B).
Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred. It's calculated as P(A|B) = P(Aā©B) Ć· P(B). It's used in medical testing, weather forecasting, and Bayesian statistics.
nCr (n choose r) is the number of ways to select r items from a group of n items when the order of selection doesn't matter. The formula is nCr = n! Ć· (r! Ć (nār)!). For example, 5C2 = 5! Ć· (2! Ć 3!) = 10.
A factorial (n!) is the product of all positive integers from 1 to n. For example, 5! = 5 Ć 4 Ć 3 Ć 2 Ć 1 = 120. By definition, 0! = 1. Factorials grow very quickly ā 20! is already over 2 quintillion.