GRE Probability Exam Questions Practice Questions with Answers

Calculate the likelihood of a specific event occurring by dividing the number of favorable outcomes by the total number of possible outcomes. Probability is a fundamental pillar of the GRE Quantitative Reasoning section, testing your ability to analyze randomness and uncertainty. Whether you are dealing with independent events, mutually exclusive scenarios, or complex combinations, mastering GRE Probability Exam Questions is essential for achieving a high score. According to the Educational Testing Service (ETS), data analysis—which includes probability—makes up a significant portion of the math section. This guide provides a deep dive into the rules of probability, ranging from basic fractions to advanced counting principles, ensuring you are prepared for any challenge on test day.

Concept Explanation

Probability is the mathematical measure of the chance that a specific event will occur, expressed as a number between 0 and 1 inclusive. A probability of 0 means the event is impossible, while a probability of 1 means it is certain. For any event $E$, the formula is:

$$P(E) = \frac{ \text{Number of favorable outcomes}}{ \text{Total number of possible outcomes}}$$

To succeed on the GRE, you must understand several key rules:

• Complementary Events: The probability that an event does not occur is $1 - P(E)$. This is often easier to calculate when the question asks for "at least one."
• Addition Rule (OR): For mutually exclusive events (events that cannot happen at the same time), $P(A \text{ or } B) = P(A) + P(B)$. If they are not mutually exclusive, $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.
• Multiplication Rule (AND): For independent events, the probability of both occurring is $P(A \text{ and } B) = P(A) \times P(B)$.
• Independent vs. Dependent: Events are independent if the outcome of one does not affect the other (like rolling two dice). They are dependent if the first outcome changes the total pool (like drawing cards without replacement).

When preparing for the quantitative section, utilizing a comprehensive GRE Prep resource can help you identify which probability patterns appear most frequently. Many students find that combining probability with counting methods like permutations and combinations is the most difficult aspect of the exam.

Solved Examples

Example 1: Independent Events
A fair six-sided die is rolled twice. What is the probability that both rolls result in a number greater than 4?

1. Identify the favorable outcomes for a single roll: The numbers greater than 4 are 5 and 6. So, there are 2 favorable outcomes.
2. Calculate the probability for one roll: $P(>4) = \frac{2}{6} = \frac{1}{3}$.
3. Since the rolls are independent, multiply the probabilities: $\frac{1}{3} \times \frac{1}{3} = \frac{1}{9}$.
4. Final Answer: $\frac{1}{9}$.

Example 2: The Complement Rule
If the probability of it raining tomorrow is 0.35, what is the probability that it will not rain?

1. Use the complement rule: $P( \text{not } A) = 1 - P(A)$.
2. Substitute the given value: $1 - 0.35 = 0.65$.
3. Final Answer: 0.65.

Example 3: Dependent Events (Without Replacement)
A bag 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?

1. Calculate the probability of the first marble being red: $\frac{5}{8}$.
2. Calculate the probability of the second marble being red, given one red is gone: $\frac{4}{7}$.
3. Multiply the probabilities: $\frac{5}{8} \times \frac{4}{7} = \frac{20}{56}$.
4. Simplify the fraction: $\frac{20}{56} = \frac{5}{14}$.
5. Final Answer: $\frac{5}{14}$.

Practice Questions

1. A jar contains 4 green, 6 white, and 10 blue marbles. If one marble is picked at random, what is the probability that it is NOT white?

2. Two cards are drawn from a standard deck of 52 cards without replacement. What is the probability that both cards are Aces?

3. A fair coin is flipped three times. What is the probability of getting exactly two heads?

4. Event A and Event B are independent. If $P(A) = 0.4$ and $P(B) = 0.5$, what is the probability that at least one of the events occurs?

5. In a group of 50 students, 20 are in the Chess Club, 25 are in the Debate Club, and 10 are in both. If a student is chosen at random, what is the probability they are in neither club?

6. A box contains 10 light bulbs, of which 3 are defective. If 2 bulbs are selected at random without replacement, what is the probability that at least one is defective?

7. A password consists of 3 digits (0-9). If the digits can be repeated, what is the probability that the password consists of the same three digits (e.g., 111, 222)?

8. What is the probability of rolling a sum of 7 or 11 with two fair six-sided dice?

9. A committee of 3 is to be selected from a group of 5 men and 4 women. What is the probability that the committee consists of 2 men and 1 woman?

10. If a letter is chosen at random from the word "PROBABILITY", what is the probability that it is a vowel?

Answers & Explanations

1. Answer: $\frac{7}{10}$ or 0.7
   Total marbles = $4 + 6 + 10 = 20$. White marbles = 6. Probability of white = $\frac{6}{20} = \frac{3}{10}$. Probability of NOT white = $1 - \frac{3}{10} = \frac{7}{10}$.
2. Answer: $\frac{1}{221}$
   First Ace: $\frac{4}{52} = \frac{1}{13}$. Second Ace: $\frac{3}{51} = \frac{1}{17}$. Multiply: $\frac{1}{13} \times \frac{1}{17} = \frac{1}{221}$.
3. Answer: $\frac{3}{8}$
   Total outcomes = $2^3 = 8$. The outcomes with exactly 2 heads are HHT, HTH, and THH (3 outcomes). Probability = $\frac{3}{8}$.
4. Answer: 0.7
   $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. Since independent, $P(A \text{ and } B) = 0.4 \times 0.5 = 0.2$. So, $0.4 + 0.5 - 0.2 = 0.7$.
5. Answer: $\frac{3}{10}$ or 0.3
   Using the inclusion-exclusion principle: Total in clubs = $20 + 25 - 10 = 35$. Neither club = $50 - 35 = 15$. Probability = $\frac{15}{50} = \frac{3}{10}$.
6. Answer: $\frac{8}{15}$
   Use the complement: $1 - P( \text{none defective})$. Probability both are good: $\frac{7}{10} \times \frac{6}{9} = \frac{42}{90} = \frac{7}{15}$. Probability at least one defective = $1 - \frac{7}{15} = \frac{8}{15}$.
7. Answer: $\frac{1}{100}$
   Total combinations = $10 \times 10 \times 10 = 1000$. Favorable outcomes (000, 111, ..., 999) = 10. Probability = $\frac{10}{1000} = \frac{1}{100}$.
8. Answer: $\frac{2}{9}$
   Total outcomes for two dice = 36. Sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes. Sum of 11: (5,6), (6,5) = 2 outcomes. Total favorable = $6 + 2 = 8$. Probability = $\frac{8}{36} = \frac{2}{9}$.
9. Answer: $\frac{10}{21}$
   Total ways to choose 3 from 9: $\binom{9}{3} = 84$. Ways to choose 2 men from 5 and 1 woman from 4: $\binom{5}{2} \times \binom{4}{1} = 10 \times 4 = 40$. Probability = $\frac{40}{84} = \frac{10}{21}$.
10. Answer: $\frac{4}{11}$
    The word "PROBABILITY" has 11 letters. Vowels are O, A, I, I (4 vowels). Probability = $\frac{4}{11}$.

To further sharpen your skills, you can use an AI Question Generator to create custom sets of GRE-style probability problems. This is particularly helpful for mastering the nuances between permutations and combinations in probability contexts.

1. If the probability of event X is 0.2 and the probability of event Y is 0.4, and the events are independent, what is the probability that both occur?

• A0.6
• B0.08
• C0.2
• D0.8

Frequently Asked Questions

What is the difference between independent and mutually exclusive events?

Independent events are those where the occurrence of one does not change the probability of the other, while mutually exclusive events are those that cannot happen at the same time. For example, rolling a 4 and rolling a 5 on a single die are mutually exclusive, but rolling a 4 on the first die and a 5 on the second die are independent.

How do I solve "at least one" probability questions on the GRE?

The most efficient way to solve "at least one" problems is to calculate the probability of the event never happening and subtract that from 1. This uses the complement rule, which simplifies complex calculations involving multiple successful scenarios.

Do I need to know permutations and combinations for GRE probability?

Yes, many high-level GRE probability questions require you to use combinations to find the total number of possible outcomes or favorable outcomes. You can practice these specific counting techniques using an AI Exam Simulator to get a feel for the actual test environment.

What is the probability of an event that is certain to happen?

An event that is certain to happen has a probability of 1. In the context of the GRE, this is often represented as 100% in word problems, though the quantitative section typically uses fractions or decimals.

Can a probability be negative?

No, a probability can never be negative because it represents a ratio of counts, and you cannot have a negative number of outcomes. The range for any probability is strictly between 0 and 1.