**GRE Data Analysis: Probability & Counting Methods – Complete Study Guide**

GRE Data Analysis: Probability & Counting Methods – Complete Study Guide

(For the 320+ Scorer)

What This Is

Probability and counting methods appear in ~15% of GRE Quant questions—often as standalone problems or embedded in Data Interpretation sets. Mastering these concepts lets you solve 2–3 extra questions per test, directly boosting your score. The GRE tests basic rules (AND/OR), counting principles (permutations/combinations), and conditional probability—but with tricky wording to disguise simple math.

Real GRE-Style Example:
A box contains 5 red marbles and 3 blue marbles. If two marbles are drawn at random without replacement, what is the probability that both are red? (A) 5/14 (B) 5/8 (C) 25/64 (D) 5/16 (E) 3/28

Why it’s tested: The GRE rewards structured thinking over memorization. Probability questions test whether you can: 1. Translate words into math (e.g., "without replacement" → dependent events).
2. Choose the right counting method (combinations vs. permutations).
3. Avoid traps (e.g., misapplying "OR" vs. "AND" rules).

Key Concepts & Techniques

1. Basic Probability Rules

• Formula: P(A) = (Number of favorable outcomes) / (Total possible outcomes)
• When to use: Single-event probability (e.g., rolling a die, drawing one card).
• GRE trap: Total outcomes must account for all possible results (e.g., for two dice, total outcomes = 6 × 6 = 36, not 12).

2. AND vs. OR Probability

• AND (Multiplication Rule):
• P(A and B) = P(A) × P(B|A) (if dependent) or P(A) × P(B) (if independent).
• When to use: Sequential events (e.g., drawing two cards without replacement).

• Example: Probability both marbles are red = (5/8) × (4/7) = 20/56 = 5/14.

• OR (Addition Rule):

• P(A or B) = P(A) + P(B) – P(A and B) (if events overlap).
• P(A or B) = P(A) + P(B) (if mutually exclusive).
• When to use: "At least one" scenarios (e.g., probability of rolling a 2 or a 5 on a die).

3. Complementary Probability

• Formula: P(not A) = 1 – P(A)
• When to use: "At least one" problems (e.g., probability of not getting all tails in 3 coin flips).
• Shortcut: Calculate P(all tails) = (1/2)³ = 1/8 → P(at least one head) = 1 – 1/8 = 7/8.

4. Counting Methods: Fundamental Counting Principle (FCP)

• Rule: If Event A has m outcomes and Event B has n outcomes, total outcomes = m × n.
• When to use: Independent choices (e.g., outfits: 3 shirts × 4 pants × 2 shoes = 24 combinations).
• GRE trap: FCP does not apply if choices are restricted (e.g., "no two people can sit next to each other").

5. Permutations (Order Matters)

• Formula: P(n, k) = n! / (n – k)! (arrangements of k items from n).
• When to use: Rankings, passwords, or any scenario where order matters (e.g., "How many ways can 3 people finish 1st, 2nd, 3rd in a race?").
• Example: P(5, 3) = 5! / 2! = 60.

6. Combinations (Order Doesn’t Matter)

• Formula: C(n, k) = n! / [k!(n – k)!] (groups of k items from n).
• When to use: Committees, teams, or any scenario where order is irrelevant (e.g., "How many ways to choose 3 books from 5?").
• Example: C(5, 3) = 10.
• GRE trap: C(n, k) = C(n, n–k) (e.g., C(5, 2) = C(5, 3)). Use this to simplify calculations.

7. Conditional Probability

• Formula: P(A|B) = P(A and B) / P(B) (probability of A given B occurred).
• When to use: "If" scenarios (e.g., "If a card drawn is red, what’s the probability it’s a heart?").
• Example: In a deck, P(Red and Heart) = 13/52, P(Red) = 26/52 → P(Heart|Red) = (13/52) / (26/52) = 1/2.

8. Probability Trees

• When to use: Multi-step dependent events (e.g., drawing marbles without replacement).
• How to draw:
• First branch: Probabilities of first event.
• Second branch: Probabilities of second event given the first.
• Multiply along branches to find joint probabilities.

Step-by-Step Strategy

For Every Probability/Counting Question:

1. Identify the type of problem:
2. Single event? → Basic probability.
3. Sequential events? → AND rule (multiplication).
4. "At least one"? → Complementary probability.
5. Order matters? → Permutations.
6. Order irrelevant? → Combinations.

7. "Given that"? → Conditional probability.

8. List total possible outcomes:

9. For probability: Denominator = total outcomes.

10. For counting: Use FCP, permutations, or combinations.

11. List favorable outcomes:

12. Break into cases if needed (e.g., "exactly one red marble" = red then blue + blue then red).

13. Apply the correct formula:

14. Write out the formula before plugging in numbers to avoid mistakes.

15. Simplify and eliminate:

16. Reduce fractions early.

17. Compare answer choices to your result (e.g., if you get 5/14, eliminate choices like 5/8).

18. Check for traps:

19. Did you account for replacement?
20. Did you confuse permutations/combinations?
21. Did you misapply AND/OR?

Fully Worked GRE-Style Example

Question:
A committee of 4 is to be selected from 6 women and 3 men. What is the probability that the committee includes exactly 2 women?

Step 1: Identify the type.
- Order doesn’t matter → combinations.
- Probability of a specific group → favorable outcomes / total outcomes.

Step 2: Total possible outcomes.
- Total ways to choose 4 people from 9: C(9, 4) = 126.

Step 3: Favorable outcomes.
- Choose 2 women from 6: C(6, 2) = 15.
- Choose 2 men from 3: C(3, 2) = 3.
- Total favorable = 15 × 3 = 45.

Step 4: Apply probability formula.
- P(exactly 2 women) = 45 / 126 = 5/14.

Step 5: Match to answer choices.
- (A) 5/14 → Correct.

Step 6: Check for traps.
- Did you use permutations? No (order doesn’t matter).
- Did you miscount? C(6, 2) is 15, not 30.
- Did you forget to multiply? 15 × 3 = 45, not 18.

Common Mistakes

1. Mistake: Using permutations when order doesn’t matter (or vice versa).
2. Why it happens: Confusing "arrangements" (permutations) with "groups" (combinations).

3. Correct approach: Ask: "Does swapping two items create a new outcome?" If yes → permutations. If no → combinations.

4. Mistake: Ignoring "without replacement" in probability.

5. Why it happens: Assuming events are independent when they’re not.

6. Correct approach: For sequential events without replacement, adjust the denominator (e.g., drawing two marbles: first draw = 8 total, second draw = 7 total).

7. Mistake: Adding probabilities for non-mutually exclusive events without subtracting overlap.

8. Why it happens: Forgetting P(A or B) = P(A) + P(B) – P(A and B).

9. Correct approach: If events can occur together (e.g., drawing a red card or a king), subtract the overlap (red kings).

10. Mistake: Misapplying complementary probability.

11. Why it happens: Calculating P(at least one) directly instead of 1 – P(none).

12. Correct approach: For "at least one" problems, always use the complement (e.g., P(at least one head in 3 flips) = 1 – P(all tails)).

13. Mistake: Overcounting in combinations.

14. Why it happens: Treating C(5, 2) and C(5, 3) as different when they’re equal.
15. Correct approach: Use C(n, k) = C(n, n–k) to simplify (e.g., C(10, 8) = C(10, 2)).

GRE Traps & Timing

Traps to Watch For:

1. "At least" vs. "Exactly":
2. At least 2 women = 2 women + 3 women + 4 women.
3. Exactly 2 women = only 2 women.

4. Avoid: Calculating "at least" when the question asks for "exactly."

5. Replacement vs. No Replacement:

6. With replacement: Denominator stays the same (e.g., drawing a card, putting it back).
7. Without replacement: Denominator decreases (e.g., drawing two marbles from a bag).

8. Avoid: Assuming independence when events are dependent.

9. Order Matters vs. Doesn’t Matter:

10. Permutations: Passwords, rankings, seating arrangements.
11. Combinations: Teams, committees, groups.

12. Avoid: Using P(n, k) for a committee (order irrelevant).

13. Conditional Probability Misinterpretation:

14. P(A|B) is not the same as P(B|A).
15. Avoid: Swapping the given and the event (e.g., P(Heart|Red) ≠ P(Red|Heart)).

Timing:

• Basic probability: 45–60 seconds.
• Combinations/permutations: 60–90 seconds.
• Conditional probability: 75–90 seconds.
• Multi-step problems: 90–120 seconds.
• If stuck: Flag and return later—don’t waste time on one question.

Quick Practice

1. Question:
   A bag contains 4 green and 6 yellow marbles. If two marbles are drawn at random without replacement, what is the probability that both are yellow?
   (A) 1/3 (B) 1/2 (C) 3/10 (D) 1/5 (E) 1/6

Answer: (A) 1/3
Solution: P(Yellow and Yellow) = (6/10) × (5/9) = 30/90 = 1/3.

1. Question:
   How many different 3-letter passwords can be formed from the letters A, B, C, D if no letter is repeated?
   (A) 4 (B) 12 (C) 24 (D) 64 (E) 81

Answer: (C) 24
Solution: Order matters → permutations: P(4, 3) = 4 × 3 × 2 = 24.

Last-Minute Cram Sheet

1. Probability formula: P(A) = Favorable / Total.
2. AND rule: Multiply probabilities (adjust for dependence).
3. OR rule: Add probabilities, subtract overlap.
4. Complementary probability: P(not A) = 1 – P(A).
5. FCP: m × n for independent choices.
6. Permutations: P(n, k) = n! / (n–k)!. Use when order matters.
7. Combinations: C(n, k) = n! / [k!(n–k)!]. Use when order doesn’t matter.
8. Conditional probability: P(A|B) = P(A and B) / P(B).
9. GRE trap: "At least one" → use complement.
10. GRE trap: "Without replacement" → adjust denominator.

⚠️ Memorize these distinctions:
- P(A and B) vs. P(A or B).
- Permutations vs. combinations.
- Dependent vs. independent events.
- C(n, k) vs. P(n, k).

Final Tip: On test day, write out the formula first before plugging in numbers. This prevents careless errors!