GRE Probability Practice Test Practice Questions with Answers

Probability measures the likelihood of a specific event occurring, expressed as a ratio between 0 and 1. On the GRE, this topic often appears as a challenging component of the Quantitative Reasoning section, testing your ability to calculate odds for single events, multiple independent outcomes, and mutually exclusive scenarios. Success on a GRE Probability Practice Test requires a firm grasp of both basic fractions and complex counting principles.

Preparing for these questions involves more than just memorizing formulas; it requires logical reasoning to determine when to multiply or add probabilities. Whether you are aiming for a top-tier graduate program or simply looking to brush up on your math skills, using a comprehensive GRE Prep resource can help you identify patterns in how the ETS (Educational Testing Service) structures these problems.

Concept Explanation

Probability is the mathematical study of randomness, defined as the number of favorable outcomes divided by the total number of possible outcomes in a sample space. This relationship is expressed by the fundamental formula: $$P(E) = \frac{ \text{Number of Favorable Outcomes}}{ \text{Total Number of Possible Outcomes}}$$ For example, if you roll a six-sided die, the probability of rolling a 4 is $\frac{1}{6}$. Probabilities are always between 0 (impossible) and 1 (certainty).

Key rules you must know for the GRE include:

• The Complement Rule: The probability of an event NOT happening is $1 - P(E)$. This is often the fastest way to solve "at least one" problems.
• Independent Events (Multiplication Rule): If two events do not affect each other, the probability of both occurring is $P(A) \times P(B)$.
• Mutually Exclusive Events (Addition Rule): If two events cannot happen at the same time, the probability of either occurring is $P(A) + P(B)$.
• Combinatorics: Many probability questions require combinations or permutations to find the total number of outcomes.

Solved Examples

Use these worked examples to understand the logic required for standard GRE probability questions.

1. Example 1: Independent Events
   A bag contains 4 red marbles and 6 blue marbles. If two marbles are drawn with replacement, what is the probability that both are red?
   1. Identify the total number of marbles: $4 + 6 = 10$.
   2. Calculate the probability of drawing one red marble: $\frac{4}{10} = \frac{2}{5}$.
   3. Since the marble is replaced, the second draw is independent. Multiply the probabilities: $\frac{2}{5} \times \frac{2}{5} = \frac{4}{25}$.
   4. Final Answer: $\frac{4}{25}$ or 0.16.
2. Example 2: The Complement Rule
   The probability of rain on Monday is 0.3 and the probability of rain on Tuesday is 0.4. What is the probability that it will rain on at least one of these two days?
   1. Find the probability that it does NOT rain on Monday: $1 - 0.3 = 0.7$.
   2. Find the probability that it does NOT rain on Tuesday: $1 - 0.4 = 0.6$.
   3. Calculate the probability that it rains on neither day: $0.7 \times 0.6 = 0.42$.
   4. Subtract from 1 to find the probability of "at least one": $1 - 0.42 = 0.58$.
   5. Final Answer: 0.58.
3. Example 3: Combinations in Probability
   A committee of 2 people is to be selected from a group of 5 men and 3 women. What is the probability that both selected people are women?
   1. Find the total number of ways to choose 2 people from 8: $\binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28$.
   2. Find the number of ways to choose 2 women from 3: $\binom{3}{2} = 3$.
   3. Divide favorable outcomes by total outcomes: $\frac{3}{28}$.
   4. Final Answer: $\frac{3}{28}$.

Practice Questions

Test your knowledge with these GRE-style practice questions. For more advanced practice, you might find the AI Question Generator useful for creating custom sets.

1. A fair coin is flipped 3 times. What is the probability of getting exactly two heads?
2. In a box of 10 lightbulbs, 3 are defective. If 2 bulbs are chosen at random without replacement, what is the probability that neither is defective?
3. If the probability of event A occurring is 0.5 and the probability of event B occurring is 0.2, and events A and B are independent, what is the probability that neither event occurs?

4. A jar contains 5 red, 3 green, and 2 blue candies. If one candy is selected at random, what is the probability that it is NOT green?
5. Event X and Event Y are mutually exclusive. If $P(X) = 0.25$ and $P(Y) = 0.35$, what is $P(X \text{ or } Y)$?
6. Two standard six-sided dice are rolled. What is the probability that the sum of the numbers shown is 7?
7. A student takes a multiple-choice test with 4 questions. Each question has 5 options. If the student guesses randomly, what is the probability of getting all 4 questions correct?
8. A drawer contains 6 black socks and 4 white socks. If one sock is pulled out and then a second sock is pulled out without replacing the first, what is the probability that both socks are white?
9. The probability that a certain machine breaks down on any given day is $\frac{1}{10}$. What is the probability that the machine will NOT break down on three consecutive days?
10. If a number is chosen at random from the integers 1 to 20 inclusive, what is the probability that the number is a prime number?

Answers & Explanations

1. Answer: 3/8. The total outcomes for 3 flips are $2^3 = 8$. The favorable outcomes (exactly two heads) are HHT, HTH, and THH. Thus, $\frac{3}{8}$.
2. Answer: 7/15. The probability the first is not defective is $\frac{7}{10}$. The probability the second is not defective is $\frac{6}{9}$. Multiply: $\frac{7}{10} \times \frac{6}{9} = \frac{42}{90} = \frac{7}{15}$.
3. Answer: 0.4. $P( \text{not A}) = 0.5$ and $P( \text{not B}) = 0.8$. Since they are independent, multiply: $0.5 \times 0.8 = 0.4$.
4. Answer: 7/10. Total candies = 10. Green candies = 3. Not green = $10 - 3 = 7$. Probability = $\frac{7}{10}$.
5. Answer: 0.60. For mutually exclusive events, $P(X \text{ or } Y) = P(X) + P(Y)$. So, $0.25 + 0.35 = 0.60$.
6. Answer: 1/6. There are $6 \times 6 = 36$ total outcomes. Combinations that sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 such outcomes. $\frac{6}{36} = \frac{1}{6}$.
7. Answer: 1/625. Probability of guessing one right is $\frac{1}{5}$. For four independent questions: $(\frac{1}{5})^4 = \frac{1}{625}$.
8. Answer: 2/15. First white sock: $\frac{4}{10}$. Second white sock: $\frac{3}{9}$. Multiply: $\frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15}$.
9. Answer: 729/1000. Probability of not breaking down is $\frac{9}{10}$. For three days: $(\frac{9}{10})^3 = \frac{729}{1000}$ or 0.729.
10. Answer: 2/5. Prime numbers between 1 and 20 are 2, 3, 5, 7, 11, 13, 17, 19. There are 8 primes. $\frac{8}{20} = \frac{2}{5}$.

1. If the probability of an event occurring is \( p \), what is the probability of the event not occurring?

• A\( p - 1 \)
• B\( 1 - p \)
• C\( 1/p \)
• D\( 1 + p \)

Frequently Asked Questions

What is the difference between independent and mutually exclusive events?

Independent events are those where the outcome of one does not change the likelihood of the other, while mutually exclusive events are those that cannot happen at the same time.

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

The most efficient method is to calculate the probability of the event never happening and then subtract that value from 1.

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

Yes, many probability questions require you to use counting principles to determine the total number of possible outcomes in the sample space.

Can a probability be greater than 1?

No, a probability must be a value between 0 and 1, where 0 represents an impossible event and 1 represents a certainty.

What are the most common probability topics on the GRE?

Common topics include independent and dependent events, mutually exclusive events, basic coin/dice problems, and selecting items from a set without replacement.