Probability
High-Yield Summary
- Independent events don't affect each other's probability (multiply); dependent events do (each outcome changes the next).
- Mutually exclusive events can't happen together — their combined probability is a simple sum.
- Overlapping events require subtracting the shared probability to avoid double-counting.
- A set of events is exhaustive when it covers every possible outcome, so total probability = 1.
Core Probability Rules
P(A∩B) = P(A) × P(B) [independent] P(A∪B) = P(A) + P(B) − P(A∩B) [general union]
- P(A∩B) = Probability of both A and B occurring
- P(A∪B) = Probability of A or B (or both) occurring
- P(A) × P(B) = Product rule — only valid when A and B are independent
- If A and B are mutually exclusive, P(A∩B) = 0, so P(A∪B) simplifies to P(A) + P(B).
Independent vs. Dependent Events
| Type | Definition / example |
|---|---|
| Independent | One event's outcome doesn't affect the other's probability — e.g., flip a coin, then roll a die |
| Dependent | One event's outcome changes the other's probability — e.g., drawing cards without replacement |
Addition Rule: Mutually Exclusive vs. Overlapping
| Case | Formula |
|---|---|
| Mutually exclusive (can't co-occur) | P(A∪B) = P(A) + P(B) |
| Overlapping (can co-occur) | P(A∪B) = P(A) + P(B) − P(A∩B) |
Common MCAT Trap
- Forgetting to subtract P(A∩B) when events overlap double-counts the shared outcomes — always check whether events are mutually exclusive before adding probabilities.
- "Exhaustive" is about covering all outcomes (sums to 1), not about events being mutually exclusive — the two properties are independent of each other.
Quick Recall
20% of students play basketball, 30% play soccer, 10% play both. What % play at least one?
Drawing two cards from a deck without replacement — are the draws independent or dependent?