Basic Probability

Probability measures how likely something is, on a scale from 0 (impossible) to 1 (certain).

For equally likely outcomes, count the ones you want and divide by the total. Rolling a die, P(even) = 3/6 = 1/2.

The probability of something *not* happening is 1 minus the probability that it does. That shortcut saves a lot of counting.

For two events both happening, multiply — provided the first does not change the second. Two coin flips landing heads: ½ × ½ = ¼.

Careful with cards or marbles taken without replacement: the second probability changes because the pool shrank.

For a finite sample space with equally likely outcomes, P(E) = |E|/|S|. Probabilities satisfy 0 ≤ P(E) ≤ 1, P(S) = 1, and the complement rule P(Eᶜ) = 1 − P(E).

For unions, P(A ∪ B) = P(A) + P(B) − P(A ∩ B); the subtraction avoids double counting and vanishes only when A and B are mutually exclusive.

For intersections, the general rule is P(A ∩ B) = P(A)·P(B | A). Independence is the special case where P(B | A) = P(B), reducing it to plain multiplication. Drawing with replacement gives independence; drawing without replacement does not, which is why the second factor changes.

Mutually exclusive and independent are different properties and are frequently confused. Two events that cannot co-occur are strongly dependent: knowing one happened tells you the other did not.