Statistics › 🎲 Probability

The Addition & Multiplication Rules

How to combine chances: the "or" rule, the "and" rule, and the overlap everyone counts twice.

Intro StatisticsAP Statistics level

💡

The big idea: Most real questions combine two events: what is the chance of A OR B, or of A AND B? Two rules cover almost everything. For "or", add the two probabilities but subtract the part you counted twice. For "and", multiply — using the chance of the second event given the first. Getting the overlap and the "given" right is the whole game.

🎯 By the end, you'll be able to

Use the addition rule to find P(A or B), and know when you must subtract the overlap
Recognize disjoint (mutually exclusive) events and use the shortcut P(A) + P(B)
Use the multiplication rule P(A and B) = P(A)·P(B | A) with conditional probability
Tell independent events apart from disjoint ones, and use P(A and B) = P(A)P(B) when events are independent

📎 You should already know

Probability of a single event (a number between 0 and 1)
Reading probabilities as fractions, decimals, or percentages

Two little words: 'or' and 'and'

Almost every probability question hides one of two words. Or asks for the chance that at least one of two things happens. And asks for the chance that both happen together. Each word has its own rule, and mixing them up is the most common slip in this whole topic.

Picture two overlapping circles — a Venn diagram. Circle A holds every outcome where event A happens; circle B holds every outcome where B happens. The overlap is where both happen at once. Keep that picture in mind and both rules almost draw themselves.

🔑 The addition rule (for 'or')

The chance that A or B happens is the chance of A, plus the chance of B, minus the chance they happen together: \( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \). You subtract the overlap because adding the two whole circles counts the shared middle twice.

\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]

The addition rule. The last term removes the overlap that the first two terms count twice.

The disjoint shortcut

Sometimes two events cannot both happen: a single coin flip is not heads and tails, one card is not a heart and a spade. Such events are called disjoint or mutually exclusive. Their circles do not overlap at all, so \( P(A \text{ and } B) = 0 \) and the addition rule loses its last term, collapsing to a plain \( P(A) + P(B) \).

So the full addition rule always works — the disjoint version is just the special case where there is no overlap to subtract.

🎮 Interactive: the overlap you count twice LIVE

Adjust P(A), P(B), and the size of their overlap. Watch how simply adding P(A) and P(B) overshoots the true P(A or B) by exactly the shaded overlap — the piece the addition rule subtracts back out. Slide the overlap to zero to see disjoint events, where 'or' is just 'plus'.

✨ Why you subtract, in one picture

Pull the overlap to zero and the circles separate: or becomes plain plus. Now grow the overlap. The naive sum \( P(A) + P(B) \) climbs past the real answer, because the middle region sits inside both circles and gets added in from each side. Subtracting \( P(A \text{ and } B) \) once puts that middle back to a single, honest count.

The multiplication rule (for 'and')

For and, you multiply. But the second probability has to account for the first event already having happened. That 'given the first happened' probability is written \( P(B \mid A) \) and read 'the probability of B given A'. Multiply the chance of the first event by the chance of the second, measured after the first:

\[ P(A \text{ and } B) = P(A) \cdot P(B \mid A) \]

The general multiplication rule. P(B | A) is the chance of B once you know A has occurred.

🔑 Independent events

Two events are independent when one happening does not change the chance of the other, so \( P(B \mid A) = P(B) \). Then the rule simplifies to \( P(A \text{ and } B) = P(A) \cdot P(B) \). Separate coin flips and dice rolls are independent; drawing cards without putting them back is not, because taking a card changes what is left in the deck.

⚠️ Disjoint is not the same as independent

These two words get swapped constantly. Disjoint events cannot both happen, so learning that one occurred tells you the other did not — that is a very strong kind of dependence. Independent events can cheerfully happen together; one simply does not shift the other's chance. So if two events with nonzero probability are disjoint, they are not independent.

📝 Worked example: In a survey group, 60 percent play a sport, 30 percent play a musical instrument, and 20 percent do both. Pick one person at random. What is the chance they play a sport or an instrument?

Name the pieces: \( P(\text{sport}) = 0.60 \), \( P(\text{instrument}) = 0.30 \), and the overlap \( P(\text{both}) = 0.20 \).
These are not disjoint — 20 percent do both — so the overlap term matters. Use the full addition rule, not the shortcut.
Substitute: \( P(\text{sport or instrument}) = 0.60 + 0.30 - 0.20 \).

✓ 0.70, or 70 percent. Adding 0.60 and 0.30 alone gives 0.90, but that double-counts the 20 percent who do both — subtracting the overlap brings it down to the correct 0.70.

Check your understanding

1. You draw one card from a standard 52-card deck. What is the probability it is a King or a Heart?

Kings: 4/52. Hearts: 13/52. The King of Hearts sits in both, so subtract it once: 4/52 + 13/52 - 1/52 = 16/52.

2. Two events are mutually exclusive (disjoint). Which statement is always true?

Disjoint events cannot occur together, so P(A and B) = 0. The addition rule then loses its last term, leaving P(A) + P(B).

3. You flip a fair coin twice, and the flips are independent. What is the probability of getting heads both times?

Independent events multiply: P(H and H) = P(H)·P(H) = 1/2 · 1/2 = 1/4.

4. You draw two cards from a deck without replacing the first. Which expression gives the probability that both are aces?

Use P(A and B) = P(A)·P(B | A). After one ace is removed, 3 aces remain among 51 cards, so the chance is 4/52 · 3/51.

✅ Key takeaways

'Or' uses the addition rule: P(A or B) = P(A) + P(B) − P(A and B); subtract the overlap so it is not counted twice.
Disjoint (mutually exclusive) events cannot happen together, so P(A and B) = 0 and P(A or B) = P(A) + P(B).
'And' uses the multiplication rule: P(A and B) = P(A)·P(B | A), where P(B | A) accounts for A already having happened.
Independent events do not affect each other, so P(B | A) = P(B) and P(A and B) = P(A)·P(B).
Disjoint and independent are different ideas: disjoint events with nonzero probability are strongly dependent, not independent.