Addition and Multiplication Rules

These two rules combine probabilities of separate events. The addition rule answers “A or B”, and the multiplication rule answers “A and B”.

Addition Rule

P(A or B) = P(A) + P(B) − P(A and B)

• P(A or B): the probability that at least one of the two events happens.
• P(A and B): the probability that both happen together. It is subtracted because adding P(A) and P(B) counts the overlap twice.

If A and B can never happen together (mutually exclusive), the overlap is zero, and the rule shrinks to P(A) + P(B).

Multiplication Rule

P(A and B) = P(A) × P(B), for independent events

• Independent: one event happening does not change the probability of the other.
• P(A) × P(B): the chance of A, scaled by the chance of B.

Example: Two servers run independently. Each has a 5% chance of going down on a given day. Find the probability that both go down, and the probability that at least one goes down.

1. Both down: 0.05 × 0.05 = 0.0025
2. At least one down: 0.05 + 0.05 − 0.0025 = 0.0975

So there is a 0.25% chance of a full outage and a 9.75% chance of some outage. Notice that 9.75% is slightly less than the 10% you would get by adding the two chances without the correction.