Probability rules trip students up because two pairs of words sound interchangeable but are not. "Mutually exclusive" and "independent" describe different relationships, and the addition rule is for "or" while the multiplication rule is for "and." Get those two distinctions right and almost every introductory probability problem becomes routine.
The Two Big Distinctions
Before any formula, separate the two key vocabulary pairs.
Mutually exclusive vs. not mutually exclusive. Two events are mutually exclusive (or disjoint) if they cannot both happen on the same trial. Rolling a 3 and rolling a 5 on one die are mutually exclusive. Drawing a King and drawing a Heart on one card are not — the King of Hearts is both.
Independent vs. dependent. Two events are independent if the outcome of one has no effect on the probability of the other. Flipping a coin twice gives independent events. Drawing two cards without replacement from a deck gives dependent events — the first card changes what is left for the second.
A trap: mutually exclusive and independent are sometimes confused, but they are almost opposites. If two events are mutually exclusive and both have nonzero probability, they cannot be independent — knowing one happened tells you the other definitely did not. Keep these two distinctions on separate axes in your head.
The Addition Rule (for "Or")
Use the addition rule when the question asks for the probability that A or B happens (the union, A ∪ B).
The general addition rule is:
P(A or B) = P(A) + P(B) − P(A and B)
The subtraction undoes double-counting: outcomes in both A and B were added once for A and once for B, so subtract them once.
When A and B are mutually exclusive, P(A and B) = 0, so the formula collapses to the special case:
P(A or B) = P(A) + P(B)
Worked example. Draw one card from a standard 52-card deck. What is the probability of a King or a Heart?
P(King) = 4/52. P(Heart) = 13/52. P(King and Heart) = 1/52 (the King of Hearts). So P(King or Heart) = 4/52 + 13/52 − 1/52 = 16/52 ≈ 0.308.
If you had instead been asked for "King or Queen," those events are mutually exclusive — no single card is both — so P = 4/52 + 4/52 = 8/52 ≈ 0.154 with no subtraction.
The Multiplication Rule (for "And")
Use the multiplication rule when the question asks for the probability that A and B both happen (the intersection, A ∩ B).
The general multiplication rule is:
P(A and B) = P(A) × P(B | A)
where P(B | A) is the conditional probability of B given that A occurred.
When A and B are independent, knowing A happened does not change the probability of B, so P(B | A) = P(B) and the formula collapses to the special case:
P(A and B) = P(A) × P(B)
Worked example, independent. Flip a fair coin twice. The two flips are independent (each is a fresh trial), so P(heads then heads) = 1/2 × 1/2 = 1/4.
Worked example, dependent. Draw two cards without replacement from a 52-card deck. Probability both are Hearts?
P(first Heart) = 13/52. After one Heart is drawn, 12 Hearts remain among 51 cards. P(second Heart | first Heart) = 12/51. So P(both Hearts) = 13/52 × 12/51 = 156/2652 ≈ 0.0588.
Notice how the second probability is no longer 13/52. Dependence changes the conditional probability, and ignoring that change is the most common error on multiplication problems.
A Decision Tree for Picking the Right Rule
When a problem hands you a probability question, run through these in order:
- "Or" in the question? Addition rule. Are A and B mutually exclusive? If yes, just add. If no, subtract P(A and B).
- "And" in the question? Multiplication rule. Are A and B independent? If yes, multiply. If no, use the conditional probability for the second event.
- "Not" or "at least one" in the question? Use the complement: P(not A) = 1 − P(A), and P(at least one) = 1 − P(none).
The complement trick is worth a separate look. "At least one Heart in 4 draws" is messy by the addition rule but trivial by complement: P(at least one Heart) = 1 − P(no Hearts in any draw). For independent draws with replacement, that is 1 − (39/52)⁴ ≈ 0.683.
Conditional Probability and Independence, Formally
The conditional probability of B given A is
P(B | A) = P(A and B) / P(A)
(provided P(A) > 0). This is the multiplication rule rearranged. Two events are independent exactly when P(B | A) = P(B), or equivalently when P(A and B) = P(A) × P(B). That equation is the formal definition of independence, and a useful check: compute both sides; if they match, the events are independent in your model.
A worked check: in a class of 100 students, 40 take Stats and 30 take Econ; 12 take both. P(Stats) = 0.40, P(Econ) = 0.30, P(Stats and Econ) = 0.12. P(Stats) × P(Econ) = 0.40 × 0.30 = 0.12. They match exactly, so in this class taking Stats and taking Econ are independent. If the joint had been 0.20, the events would have been positively associated.
Common Mistakes to Avoid
The first is calling mutually exclusive events independent, or vice versa. They are almost always opposites — the only way to be both is if at least one event has probability zero, which is rare and uninteresting.
The second is forgetting to subtract the overlap in the general addition rule. "King or Heart" without subtracting the King of Hearts gives 17/52 instead of 16/52. The general rule is the default; the special rule (just adding) only applies when the events are genuinely mutually exclusive.
The third is assuming independence when sampling is without replacement. Each draw changes the deck, so the second draw's conditional probability is not the same as the first draw's. If the problem says "without replacement," you cannot multiply unconditional probabilities.
Getting Help
Once you are comfortable with the basic rules here, conditional probability and independence become the foundation for everything from the binomial distribution walkthrough to expected value. Browse the full Statistics & Data hub for more concept walkthroughs that build on these rules.
Conclusion
Probability rules for addition and multiplication come down to two questions: are the events mutually exclusive (controls how addition works), and are they independent (controls how multiplication works). The general forms — P(A or B) = P(A) + P(B) − P(A and B) and P(A and B) = P(A) × P(B | A) — always work; the special forms are shortcuts when the right relationship holds. Write the question's "or"/"and" structure first, then pick the rule and check the relationship.
