Probability on the GRE is less about memorizing exotic formulas and more about recognizing patterns: complements, counting, independence, conditional probability, and “at least one” questions. The following set gives you 25 GRE-style probability problems with detailed solutions, moving from quick fundamentals to trickier reasoning. Use them as a compact workout: solve first, then compare your method.

Quick reminder: probability equals favorable outcomes divided by total outcomes. For “without replacement,” probabilities change after each draw; for “with replacement,” they do not.

1. Three fair coins are tossed. What is the probability of exactly two heads?
   Outcomes: HHT, HTH, THH are favorable out of 8 total outcomes. Probability = 3/8.
2. A fair die is rolled. What is the probability of rolling greater than 4?
   Favorable outcomes are 5 and 6, so there are 2 favorable results out of 6. Probability = 1/3.
3. Two fair dice are rolled. What is the probability that the sum is 7?
   Favorable pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 out of 36 outcomes, so probability = 1/6.
4. One card is drawn from a standard deck. What is the probability it is a heart or a king?
   There are 13 hearts and 4 kings, but the king of hearts is counted twice. Favorable = 13 + 4 − 1 = 16. Probability = 4/13.
5. A bag has 5 red and 3 blue marbles. Two are drawn without replacement. What is the probability both are red?
   First red: 5/8. Second red: 4/7. Multiply: 5/8 × 4/7 = 5/14.
6. A batch has 10 items, 2 defective. If 3 are chosen, what is the probability at least one is defective?
   Use the complement: no defective means choosing all 3 from 8 good items. Probability = 1 − C(8,3)/C(10,3) = 1 − 56/120 = 8/15.
7. If A and B are independent, P(A)=0.4 and P(B)=0.7. What is P(A and B)?
   For independent events, multiply: 0.4 × 0.7 = 0.28.
8. Using the same events, what is the probability exactly one occurs?
   Exactly one means A occurs and B does not, or B occurs and A does not: 0.4(0.3) + 0.7(0.6) = 0.12 + 0.42 = 0.54.
9. If P(A and B)=0.18 and P(B)=0.30, what is P(A given B)?
   Conditional probability: P(A|B)=P(A and B)/P(B)=0.18/0.30=0.60.
10. A group has 4 women and 2 men. Two people are selected. What is the probability both are women?
    Choose 2 women from 4 and 2 people from 6: C(4,2)/C(6,2)=6/15=2/5.
11. A spinner numbered 1 through 8 is spun. What is the probability the result is even and greater than 5?
    Even numbers greater than 5 are 6 and 8. That is 2 favorable outcomes out of 8. Probability = 1/4.
12. A code uses one letter from A, B, C and one digit from 0 through 4. What is the probability the code has B or digit 3?
    Total codes = 3 × 5 = 15. With B: 5. With digit 3: 3. Overlap B3: 1. Favorable = 7, so probability = 7/15.
13. A machine succeeds with probability 0.2 each try. What is the probability the first success occurs on the fourth try?
    The first three must fail, then the fourth succeeds: 0.8³ × 0.2 = 0.1024.
14. A fair coin is tossed 5 times. What is the probability of exactly 3 heads?
    Choose which 3 tosses are heads: C(5,3)=10. Total outcomes = 32. Probability = 10/32 = 5/16.
15. A die is rolled 4 times. What is the probability of at least one 6?
    Complement: no 6 in all four rolls is (5/6)⁴. Therefore probability = 1 − (5/6)⁴.

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16. A bag has 4 red and 6 blue marbles. Two are drawn with replacement. What is the probability both are the same color?
    Same color means both red or both blue: (4/10)² + (6/10)² = 16/100 + 36/100 = 13/25.
17. A committee of 3 is chosen from 5 adults and 4 teens. What is the probability at least 2 are teens?
    Favorable = C(4,2)C(5,1)+C(4,3)=30+4=34. Total = C(9,3)=84. Probability = 34/84 = 17/42.
18. If P(A)=0.6, P(B)=0.5, and P(A or B)=0.8, what is P(A and B)?
    Use P(A or B)=P(A)+P(B)−P(A and B). Thus 0.8=0.6+0.5−x, so x=0.3.
19. An integer from 1 to 100 is selected. What is the probability it is divisible by 3 or 5?
    Multiples of 3: 33. Multiples of 5: 20. Multiples of 15: 6. Favorable = 33+20−6=47. Probability = 47/100.
20. Three cards are drawn with replacement. What is the probability all three suits are different?
    First suit can be anything. Second must differ: 3/4. Third must differ from both: 2/4. Probability = 3/8.
21. Two dice are rolled. What is the probability their product is odd?
    A product is odd only if both dice are odd. Each die has 3 odd faces out of 6, so probability = 1/2 × 1/2 = 1/4.
22. A box has 2 green, 3 yellow, and 5 black balls. What is the probability one selected ball is not black?
    Non-black balls = 2 + 3 = 5 out of 10 total. Probability = 1/2.
23. An integer from 1 to 30 is selected. What is the probability it is an odd prime?
    Odd primes are 3, 5, 7, 11, 13, 17, 19, 23, and 29: 9 numbers. Probability = 9/30 = 3/10.
24. In a survey, 70% like math, 50% like verbal, and 30% like both. What percent like neither?
    Like at least one = 70% + 50% − 30% = 90%. Therefore neither = 10%.
25. A disease affects 2% of people. A test is positive for 90% of sick people and falsely positive for 10% of healthy people. What is P(sick given positive)?
    Positive and sick = 0.02×0.90=0.018. Total positive = 0.018 + 0.98×0.10 = 0.116. Probability = 0.018/0.116 = 9/58.

One final GRE habit: after solving, ask whether your answer is reasonable. Probabilities must fall between 0 and 1, and many intimidating questions become simple once you draw a table, count carefully, or use the complement.