Probability
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). Every basic probability is a single fraction — favourable outcomes over total equally-likely outcomes — so the whole subject leans on the counting skills from the previous chapter. The GRE keeps the scenarios concrete: coins, dice, marbles in a bag, cards, and everyday selections. What it tests is whether you can tell an independent event from a dependent one, add "or" probabilities without double-counting the overlap, and reach for the complement when a direct count would be painful. The arithmetic stays light enough for the on-screen basic calculator; the points live in the setup. Get the sample space right and one non-obvious habit — computing 1 minus the "none" case for "at least one" questions — and probability becomes one of the most predictable topics on the test.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Name the sample space first — the total count of equally-likely outcomes — before counting favourable ones.
- "And" multiplies, "or" adds. Get that reflex right and half the setups fall out immediately.
- For "at least one", compute 1 − P(none). It is almost always faster than summing the cases.
- Check "with or without replacement": without replacement makes the events dependent, so the second fraction changes.
- For "or", ask "can both happen?" If yes, subtract the overlap; if no (mutually exclusive), just add.
- Fractions usually beat decimals here and are accepted in Numeric Entry boxes — leave answers like 5/14 unsimplified only if the box demands lowest terms.
- Every probability sits between 0 and 1 — an answer outside that range signals a setup error.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Adding probabilities for "and" or multiplying for "or" — reversing the two core operations.
- Forgetting to subtract the overlap in P(A or B) when the events can co-occur.
- Treating without-replacement draws as independent (using P(A)×P(B) instead of the reduced second fraction).
- Computing "at least one" the long way instead of using 1 − P(none).
- Building a sample space of outcomes that are not equally likely (e.g. using dice sums 2–12 as if equal).
- Leaving a "probability" greater than 1 or negative and not noticing the impossibility.
📈 GRE exam insight & question patterns
Quantitative Comparison — Column A: P(at least one head in 3 tosses of a fair coin). Column B: 3/4. Which is greater?
Column A. 1 − (1/2)³ = 7/8 = 0.875 > 0.75.
Concept — Two cards are drawn without replacement from a deck. Why is P(both kings) not (4/52)²?
The draws are dependent: after one king leaves, only 3 kings remain among 51 cards, so it is (4/52)(3/51).
Numeric Entry — A bag has 3 red and 5 blue marbles; one is drawn. Enter P(red) as a fraction.
3/8.
Multiple-answer — For one roll of a die, which pairs of events are mutually exclusive? (odd & even / prime & even / >4 & <3 / rolling 6 & prime)
Three pairs are mutually exclusive: odd & even, >4 {5,6} & <3 {1,2}, and rolling a 6 & prime {2,3,5} (6 is not prime). Only prime & even overlap — at the value 2.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
- Every basic probability is favourable over total equally-likely outcomes, a value from 0 to 1.
- "And" multiplies; "or" adds — the two operations you must never swap.
- Independent events keep their probabilities; dependent (without-replacement) events shrink the second fraction.
- For "or", subtract the overlap unless the events are mutually exclusive.
- The complement rule P(not E) = 1 − P(E) is the fast path, especially for "at least one" = 1 − P(none).
- Name the sample space before counting, and watch for "with/without replacement".
- Fractions are usually cleaner than decimals and are accepted in Numeric Entry boxes.
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Probability when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 8 cards |
Formula Reference Sheet
Core probability
| Basic probability | P(E) = favourable outcomes / total equally-likely outcomes |
|---|---|
| Range of a probability | 0 ≤ P(E) ≤ 1 |
| Complement | P(not E) = 1 − P(E) |
| Independent "and" | P(A and B) = P(A) × P(B) |
Combining events
| General "or" | P(A or B) = P(A) + P(B) − P(A and B) |
|---|---|
| Mutually exclusive "or" | P(A or B) = P(A) + P(B) (no overlap) |
| Dependent "and" | P(A then B) = P(A) × P(B given A) |
| "At least one" | P(at least one) = 1 − P(none) |