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DAT Quantitative Reasoning · Review sheet · Free sample
Probability & Counting
Updated 2026-10-19
Counting
| Situation | Formula | Example |
|---|
| Sequence of choices | Multiply the options | 3 shirts × 4 pants = 12 outfits |
|---|
| Arrange all n distinct items | n! | 5 books on a shelf: 120 |
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| Ordered selection (permutation) | nPr = n! ÷ (n − r)! | President and VP from 10: 10 × 9 = 90 |
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| Unordered selection (combination) | nCr = n! ÷ [r!(n − r)!] | Committee of 2 from 5: 10 |
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| Arrangements with repeated items | n! ÷ (a! b! …) | Letters of LEVEL: 5! ÷ (2! 2!) = 30 |
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| Circular arrangements | (n − 1)! | 5 people at a round table: 24 |
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- Ask: does order matter? Yes → permutation. No → combination.
Probability rules
| Rule | Formula |
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| Basic | P = favorable outcomes ÷ total equally likely outcomes |
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| Complement | P(not A) = 1 − P(A) |
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| At least one | 1 − P(none) |
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| Or (general) | P(A or B) = P(A) + P(B) − P(A and B) |
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| And (independent) | P(A and B) = P(A) × P(B) |
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| And (dependent, without replacement) | P(A) × P(B | A) |
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| Conditional | P(A | B) = P(A and B) ÷ P(B) |
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| Exactly k successes in n trials | nCk × pᵏ × (1 − p)ⁿ⁻ᵏ |
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| Expected value | Σ (value × probability) |
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- Two dice: 36 outcomes. Sum of 7 is most likely (6 ways, 1/6). Standard deck: 52 cards, 4 suits, 13 ranks, 12 face cards.
⚠️ Common traps
- Using a permutation when order does not matter.
- Adding probabilities of events that can happen together without subtracting the overlap.
- Forgetting to reduce the total when drawing without replacement.
- Computing 'at least one' by adding cases instead of using the complement.
Mini-quiz
Pick an answer to see right away whether it is correct and why.
How many ways can a 2-person committee be chosen from 5 people?
How many distinct arrangements of the letters in LEVEL are there?
A fair coin is flipped 3 times. What is the probability of at least one head?
Two fair dice are rolled. What is the probability the sum is 7?
Two cards are drawn from a standard deck without replacement. What is the probability both are aces?
0 of 5 answered
Answer key: 1 A · 2 B · 3 D · 4 B · 5 B
