Step by Step
AND
P(A and B) — multiply
P(A and B) = P(A) × P(B|A). If A and B are independent, this simplifies to P(A) × P(B).
OR
P(A or B) — add, then subtract overlap
P(A or B) = P(A) + P(B) − P(A and B). If A and B are mutually exclusive (they can't both happen), this simplifies to just P(A) + P(B), since there's no overlap to subtract.
Indep
Independence vs. mutual exclusivity — don't confuse them
Independent events can still both occur together (their occurrence just doesn't affect each other's probability). Mutually exclusive events can never both occur together at all. These are different concepts, and mixing them up leads to incorrect formulas.
Key
The core memory anchor
AND means multiply; OR means add and then subtract the overlap (unless the events are mutually exclusive, in which case there's no overlap to subtract).
Applied Walkthrough
1
A card is drawn from a standard deck. Find P(drawing a King OR a Heart).
2
Since a King and a Heart aren't mutually exclusive (the King of Hearts is both), use the full OR formula: P(King) + P(Heart) − P(King and Heart).
3
P(King) = 4/52, P(Heart) = 13/52, P(King and Heart) = 1/52 (just the King of Hearts).
4
P(King or Heart) = 4/52 + 13/52 − 1/52 = 16/52 ≈ 0.308 — the overlap (King of Hearts) was subtracted to avoid double-counting it.
Exam Application
Exams test whether you correctly apply the AND (multiply) versus OR (add, subtract overlap) rules, and whether you correctly distinguish independence from mutual exclusivity when deciding whether to simplify the formulas.
⚠ Common Trap
The most common trap is forgetting to subtract the overlap in an OR calculation when the two events aren't mutually exclusive — this causes the overlap to be double-counted, inflating the probability.
✓ Quick Self-Check
1. What is the formula for P(A and B)?
P(A) × P(B|A), or P(A) × P(B) if independent.
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2. What is the formula for P(A or B)?
P(A) + P(B) − P(A and B), or just P(A) + P(B) if mutually exclusive.
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3. Find P(King or Heart) when drawing a card from a standard deck.
16/52 ≈ 0.308.
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4. What's the difference between independent events and mutually exclusive events?
Independent events can still both occur (their probabilities just don't affect each other); mutually exclusive events can never both occur together at all.
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5. What is the core memory anchor for the AND/OR rules?
AND means multiply; OR means add and subtract the overlap.
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