How probability is calculated
For equally likely outcomes, probability is simply the ratio of favourable outcomes to all outcomes:
P = favourable ÷ total
The complementary rule — P(not E) = 1 − P(E) — is the workhorse shortcut: it is often easier to find the probability of something failing than succeeding, so compute the easy one and subtract from 1. Odds in favour are the ratio P(E) ÷ P(not E), which is a different number from the probability itself (odds of 3 to 1 mean P = 0.75, not 0.3) and the form used in betting and actuarial work.
Worked example
Rolling a 3 on a fair six-sided die: P = 1 ÷ 6 ≈ 0.1667 = 16.67%, or 1/6 in fraction form. The complement — not rolling a 3 — is 1 − 0.1667 = 0.8333 (83.33%), and the odds in favour are 1 : 5. Note how the percentage looks "small" while the odds 1 to 5 sound "big"; the same information, two conventions, and mixing them up is one of the commonest probability errors.
Independent events and combinations
For independent events, multiply: the probability of two heads in two fair flips is 0.5 × 0.5 = 0.25, not 0.5. For "at least one", use the complement — P(at least one) = 1 − P(none) — because direct enumeration gets messy. When order does not matter but selection does, use combinations n! ÷ (k!(n−k)!): the number of ways to draw 3 cards from a 52-card deck is 22,100, whereas permutations (order matters, 52 × 51 × 50) give 132,600.