Probability fundamentals

P(event) = (favorable outcomes) / (total outcomes). For a fair die, P(rolling a 4) = 1/6. P(rolling an even number) = 3/6 = 1/2.

When the result of one event doesn't affect another, multiply probabilities: P(A and B) = P(A) × P(B). Example: P(two coins both heads) = 1/2 × 1/2 = 1/4.

If you draw something and don't put it back, the next probability changes. Example: bag of 5 marbles, 3 red. P(first red) = 3/5. P(second red | first red) = 2/4. Multiply for both red: 3/5 × 2/4 = 3/10.

P(not event) = 1 − P(event). Sometimes easier to compute the complement. P(at least one heads in 3 flips) = 1 − P(no heads) = 1 − (1/2)³ = 7/8.