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Calculate basic probability, P(A and B), P(A or B), conditional probability, combinations, permutations, coin flip, and dice roll probabilities.
Probability Formulas:
P(A or B) = P(A) + P(B) - P(A and B)
P(A | B) = P(A and B) / P(B)
nCr = n! / (r! × (n-r)!)
nPr = n! / (n-r)!
Combinations (nCr) count ways to choose r items from n where order does not matter. Permutations (nPr) count ways where order does matter. For example, choosing 3 from 10: nCr = 120, nPr = 720.
P(A|B) is the probability of A given that B has occurred: P(A|B) = P(A and B) / P(B). It adjusts the probability of A based on knowledge that B happened.
P(A or B) is the probability that at least one of A or B occurs. Using the addition rule: P(A or B) = P(A) + P(B) - P(A and B). The subtraction removes double-counting the overlap.
Two events are independent if the occurrence of one does not affect the other. For independent events: P(A and B) = P(A) × P(B). Coin flips are independent; each flip has no memory of prior results.
The probability of exactly k heads in n fair coin flips uses the binomial formula: P = C(n,k) × (0.5)^n. C(n,k) counts the number of ways to arrange k heads among n flips.