Calculate probabilities for single events, combined independent events, and count combinations or permutations. Includes full formula breakdowns.
Assumes A and B are independent events.
Single Event: P = favorable outcomes / total outcomes
P(A and B) — Independent: P(A∩B) = P(A) × P(B)
P(A or B): P(A∪B) = P(A) + P(B) − P(A∩B)
Complement: P(not A) = 1 − P(A)
Combinations nCr: n! / (r! × (n−r)!) — order does NOT matter
Permutations nPr: n! / (n−r)! — order DOES matter
Combinations (nCr) count the number of ways to choose r items from n when the order does not matter. For example, choosing 3 people from a group of 10 for a committee: 10C3 = 120. Permutations (nPr) count arrangements where order matters. For example, 3 people competing for 1st, 2nd, and 3rd place from a group of 10: 10P3 = 720. Permutations are always greater than or equal to combinations for the same n and r.
Two events are independent if the occurrence of one does not affect the probability of the other. For example, flipping a coin and rolling a die are independent — the coin result doesn't change the die outcome. For independent events, P(A and B) = P(A) × P(B). In contrast, dependent events affect each other, such as drawing two cards without replacement from a deck.