Probability Rules

You’ll be able to

Apply the complement, addition, and multiplication rules
Distinguish mutually exclusive events from independent events
Use the general addition rule to avoid double-counting

The basics: probabilities live in [0, 1]

A probability is a long-run relative frequency: over many repetitions, the fraction of times an event occurs. Every probability is between 0 and 1, and the probabilities of all outcomes in a sample space sum to 1. The complement rule follows immediately: P(not A) = 1 − P(A). Complements are often the fastest route — "at least one" problems are usually easiest as 1 − P(none).

Addition: mutually exclusive vs. overlapping

Two events are mutually exclusive (disjoint) if they cannot both happen; then P(A or B) = P(A) + P(B). If they can overlap, you must subtract the double-counted middle: the general addition rule is P(A or B) = P(A) + P(B) − P(A and B). Forgetting to subtract the overlap is the most common addition error.

Multiplication and independence

Two events are independent if knowing one occurred does not change the probability of the other. For independent events, P(A and B) = P(A)·P(B). In general, P(A and B) = P(A)·P(B | A), where P(B | A) is the conditional probability of B given A. Independent is not the same as mutually exclusive — in fact two events with positive probability that are mutually exclusive are necessarily dependent.

Core probability rules

P(not A) = 1 − P(A) · P(A or B) = P(A) + P(B) − P(A and B) · P(A and B) = P(A)·P(B|A)

The general addition rule reduces to P(A)+P(B) when A and B are disjoint. The multiplication rule reduces to P(A)·P(B) when A and B are independent.

Worked example

A single card is drawn from a standard 52-card deck. Find P(the card is a King or a Heart).

1.P(King) = 4/52 (four kings).
2.P(Heart) = 13/52 (thirteen hearts).
3.These overlap: the King of Hearts is both, so P(King and Heart) = 1/52.
4.Apply the general addition rule: P(King or Heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.

Answer: P(King or Heart) = 4/52 + 13/52 − 1/52 = 16/52 ≈ 0.308. Subtracting the King of Hearts once prevents double-counting.

Checkpoint

The probability it rains on a given day is 0.30. Assuming days are independent, what is the probability it rains on at least one of the next two days?

Tip

For "at least one" questions, go to the complement: P(at least one) = 1 − P(none). Multiplying the "none" probabilities is almost always faster and less error-prone than adding up every "one or more" case.

Checkpoint

Events A and B are mutually exclusive, with P(A) = 0.4 and P(B) = 0.3, both positive. Are A and B independent?

Answer the 2 checkpoints as you read.