GRE Probability Set 1 Practice Questions with Answers

Concept Explanation

Probability is a numerical measure of the likelihood that a specific event will occur, ranging from 0 (impossible) to 1 (certain). In the context of GRE Probability Set 1, the most fundamental formula is the ratio of favorable outcomes to the total number of possible outcomes in a sample space. This is expressed as:

$$P(E) = \frac{ \text{Number of Favorable Outcomes}}{ \text{Total Number of Possible Outcomes}}$$

To succeed on the GRE, you must distinguish between independent events, where the outcome of one does not affect the other, and dependent events. For independent events $A$ and $B$, the probability of both occurring is $P(A \text{ and } B) = P(A) \times P(B)$. Conversely, for mutually exclusive events (events that cannot happen at the same time), the probability of either occurring is $P(A \text{ or } B) = P(A) + P(B)$. Understanding the complement rule is also vital: the probability of an event not happening is $1 - P(E)$. This is often the fastest way to solve "at least one" problems. For comprehensive guidance on all math topics, visit our GRE Prep hub.

Solved Examples

1. Example 1: Single Die Roll
   What is the probability of rolling a prime number on a standard six-sided die?
   
   1. Identify the total outcomes: {1, 2, 3, 4, 5, 6}. Total = 6.
   2. Identify the favorable outcomes (prime numbers): {2, 3, 5}. Favorable = 3.
   3. Apply the formula: $P = \frac{3}{6} = \frac{1}{2}$.
2. Example 2: 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. Probability of red on first draw: $\frac{4}{10} = \frac{2}{5}$.
   2. Because of replacement, the probability remains the same for the second draw: $\frac{2}{5}$.
   3. Multiply the probabilities: $\frac{2}{5} \times \frac{2}{5} = \frac{4}{25}$.
3. Example 3: The Complement Rule
   If the probability of rain tomorrow is 0.15, what is the probability that it does not rain?
   
   1. Identify the probability of the event: $P( \text{Rain}) = 0.15$.
   2. Use the complement formula: $P( \text{No Rain}) = 1 - P( \text{Rain})$.
   3. Calculate: $1 - 0.15 = 0.85$.

Practice Questions

1. A jar contains 5 clear marbles, 3 green marbles, and 2 purple marbles. If one marble is chosen at random, what is the probability that it is not green?

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

3. In a group of 20 students, 12 are taking Spanish, 8 are taking French, and 4 are taking both. If a student is picked at random, what is the probability they are taking neither Spanish nor French?

4. Two fair six-sided dice are rolled simultaneously. What is the probability that the sum of the two numbers shown is 9?

5. Box A contains 3 red and 2 white chips. Box B contains 4 red and 1 white chip. If one chip is drawn from each box, what is the probability that both chips are white?

6. A committee of 2 people is to be selected from a group of 5 men and 3 women. What is the probability that the committee consists of 2 women?

7. If a digit is chosen at random from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}, what is the probability that the digit is a multiple of 3 or an even number?

8. A spinner is divided into 8 equal sections numbered 1 through 8. If the spinner is spun twice, what is the probability that the first spin lands on an even number and the second spin lands on a number greater than 5?

Answers & Explanations

1. Answer: 7/10
   Total marbles = $5 + 3 + 2 = 10$. Marbles that are NOT green = $5 (clear) + 2 (purple) = 7$. Probability = $7/10$.
2. Answer: 3/8
   Total outcomes for 3 flips = $2^3 = 8$. The outcomes with exactly two heads are {HHT, HTH, THH}. Total favorable = 3. Probability = $3/8$.
3. Answer: 1/5
   Using the Principle of Inclusion-Exclusion: $P(S \cup F) = P(S) + P(F) - P(S \cap F) = 12 + 8 - 4 = 16$. Students taking neither = $20 - 16 = 4$. Probability = $4/20 = 1/5$.
4. Answer: 1/9
   Total outcomes = $6 \times 6 = 36$. Favorable outcomes for a sum of 9: (3,6), (4,5), (5,4), (6,3). Total favorable = 4. Probability = $4/36 = 1/9$.
5. Answer: 2/25
   P(White from A) = $2/5$. P(White from B) = $1/5$. Since these are independent: $2/5 \times 1/5 = 2/25$.
6. Answer: 3/28
   Total ways to choose 2 from 8: $\binom{8}{2} = \frac{8 \times 7}{2} = 28$. Ways to choose 2 women from 3: $\binom{3}{2} = 3$. Probability = $3/28$.
7. Answer: 2/3
   Multiples of 3: {3, 6, 9}. Even numbers: {2, 4, 6, 8}. Combined set (union): {2, 3, 4, 6, 8, 9}. Total favorable = 6. Total outcomes = 9. Probability = $6/9 = 2/3$.
8. Answer: 3/16
   P(Even on first) = $4/8 = 1/2$. P(Greater than 5 on second: 6, 7, 8) = $3/8$. Probability = $1/2 \times 3/8 = 3/16$.

1. If the probability of event A occurring is 0.4 and the probability of event B occurring is 0.5, and they are independent, what is the probability that neither occurs?

• A0.2
• B0.3
• C0.7
• D0.9

Frequently Asked Questions

What is the difference between independent and dependent events in GRE probability?

Independent events are those where the outcome of the first event has no impact on the outcome of the second, such as flipping a coin twice. Dependent events occur when the first outcome changes the total pool or conditions for the second, such as drawing cards from a deck without replacement.

How do I calculate the probability of "at least one" event occurring?

The most efficient way to solve "at least one" problems is to calculate the probability of the event never occurring and subtracting that from 1. This uses the complement rule to simplify what would otherwise be a complex multi-step calculation.

What is the total sum of all possible probabilities in a sample space?

The sum of the probabilities of all possible mutually exclusive outcomes in a sample space must always equal exactly 1. This reflects the certainty that one of the possible outcomes within that defined space must occur.

Can a probability be negative or greater than 1?

No, a probability must always be a value between 0 and 1, inclusive. A probability of 0 means the event is impossible, while a probability of 1 means the event is absolutely certain to happen.

When should I add probabilities versus multiplying them?

You should add probabilities when you are looking for the likelihood of either one event OR another occurring (mutually exclusive). You should multiply probabilities when you are looking for the likelihood of one event AND another occurring (independent events).