Easy GRE Conditional Probability Questions Practice Questions

Concept Explanation

Conditional probability is the likelihood of an event occurring given that another event has already happened. In the context of the Quantitative Reasoning section of the GRE, this concept requires you to restrict the sample space to only those outcomes that satisfy the given condition. The fundamental formula for conditional probability is:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

In this formula, $P(A|B)$ represents the probability of event A occurring given that B is true. The term $P(A \cap B)$ is the probability of both events happening simultaneously, and $P(B)$ is the probability of the condition itself. For many Easy GRE Conditional Probability Questions, you can often solve the problem by simply identifying the new "total" (the denominator) and the "favorable outcomes" within that new total (the numerator). This approach is frequently used in GRE Reading Passage Questions where data interpretation is required. Understanding how to narrow your focus to a specific subset of data is a vital skill for the GRE Prep journey.

According to the Khan Academy, conditional probability is essentially about updating your beliefs based on new information. On the GRE, this often manifests as problems involving marbles in a bag, dice rolls, or data tables. If you are looking for more ways to test your quantitative skills, you might explore GRE Practice Questions with Answers to see how these logic patterns reappear across different topics.

Solved Examples

1. Example 1: The Dice Roll. A fair six-sided die is rolled. What is the probability that the result is a 4, given that the result is an even number?
   1. Identify the condition: The result must be even. The set of even numbers is $\{2, 4, 6\}$.
   2. Identify the favorable outcome: Within the set $\{2, 4, 6\}$, only the number 4 satisfies the condition.
   3. Calculate the probability: There is 1 favorable outcome out of 3 possible outcomes. $$\text{Probability} = \frac{1}{3}$$
2. Example 2: Colored Marbles. A bag contains 5 red marbles and 5 blue marbles. Two marbles are drawn without replacement. If the first marble drawn is red, what is the probability that the second marble drawn is also red?
   1. Identify the new sample space: Since one red marble was removed, there are now 9 marbles left in the bag.
   2. Identify the remaining favorable outcomes: There were 5 red marbles; after removing one, 4 red marbles remain.
   3. Calculate the probability: $$\text{Probability} = \frac{4}{9}$$
3. Example 3: Data Tables. In a class of 20 students, 12 are female and 8 are male. Of the females, 3 are biology majors. If a student is chosen at random and found to be female, what is the probability she is a biology major?
   1. Identify the restricted group: The condition is that the student is female. Our total is now 12.
   2. Identify the favorable outcome: Among those 12 females, 3 are biology majors.
   3. Calculate the probability: $$\text{Probability} = \frac{3}{12} = \frac{1}{4} = 0.25$$

Practice Questions

1. A card is drawn from a standard deck of 52 cards. Given that the card is a heart, what is the probability that it is the Queen of Hearts?

2. A fair coin is tossed three times. Given that at least two of the tosses resulted in Heads, what is the probability that all three tosses resulted in Heads?

3. In a survey of 100 people, 60 like coffee, 40 like tea, and 20 like both. If a person is selected at random and likes coffee, what is the probability they also like tea?

4. A jar contains 4 green candies and 6 yellow candies. If two candies are picked without replacement and the first is yellow, what is the probability the second is green?

5. Two fair dice are rolled. If the sum of the dice is 8, what is the probability that at least one of the dice shows a 5?

6. A group of 50 athletes includes 30 runners and 20 swimmers. Ten of the runners also swim. If an athlete is chosen at random and is a runner, what is the probability they are also a swimmer?

7. A box contains 10 light bulbs, 3 of which are defective. If a bulb is tested and found to be defective, and then a second bulb is tested, what is the probability the second bulb is also defective?

8. Given that a randomly selected integer between 1 and 10 (inclusive) is prime, what is the probability that it is an odd number?

Answers & Explanations

1. Answer: 1/13. There are 13 hearts in a deck. The condition restricts our sample space to these 13 cards. Only one of these is the Queen of Hearts. Thus, the probability is $\frac{1}{13}$.
2. Answer: 1/4. The sample space for three tosses is 8 outcomes. The condition "at least two Heads" includes {HHH, HHT, HTH, THH}. There are 4 such outcomes. Only one of these is HHH. Therefore, the probability is $\frac{1}{4}$.
3. Answer: 1/3. The condition is that the person likes coffee (60 people). Out of these 60, 20 also like tea. The probability is $\frac{20}{60} = \frac{1}{3}$.
4. Answer: 4/9. After one yellow candy is removed, 9 candies remain in the jar. All 4 green candies are still in the jar. The probability is $\frac{4}{9}$.
5. Answer: 2/5. The sums that equal 8 are (2,6), (3,5), (4,4), (5,3), and (6,2). There are 5 possible outcomes. Two of these contain a 5: (3,5) and (5,3). The probability is $\frac{2}{5}$.
6. Answer: 1/3. The condition is that the athlete is a runner (30 people). Among these 30 runners, 10 are also swimmers. The probability is $\frac{10}{30} = \frac{1}{10}$.
7. Answer: 2/9. After the first defective bulb is found, 9 bulbs remain, and only 2 of them are defective. The probability is $\frac{2}{9}$.
8. Answer: 3/4. The prime numbers between 1 and 10 are 2, 3, 5, and 7. There are 4 primes. The odd primes are 3, 5, and 7. Thus, the probability is $\frac{3}{4}$.

1. A bag has 3 red and 7 white balls. If you draw one ball and it is white, and then draw a second ball without replacement, what is the probability the second is red?

• A3/10
• B3/9
• C7/9
• D1/3

Frequently Asked Questions

What is the difference between independent events and conditional probability?

Independent events are those where the occurrence of one does not affect the probability of the other, meaning $P(A|B) = P(A)$. Conditional probability specifically measures how the probability of one event changes once we know another event has occurred.

How do you identify a conditional probability question on the GRE?

Look for keywords like "if," "given that," "knowing that," or "of those." These phrases signal that the total sample space has been restricted to a specific subgroup.

Can conditional probability be higher than the original probability?

Yes, conditional probability can be higher, lower, or the same as the original probability. For example, the probability of drawing an Ace is 4/52, but given that the card is a Spade, the probability of it being an Ace remains 1/13, which is the same.

Do I always need to use the formula $P(A|B) = P(A \cap B) / P(B)$?

No, for most Easy GRE Conditional Probability Questions, it is faster to simply count the favorable outcomes within the restricted sample space. Using the formula is more helpful for complex problems where counts aren't easily available.

What happens if the condition is impossible?

If the condition (Event B) has a probability of zero, then the conditional probability $P(A|B)$ is undefined. On the GRE, you will only encounter conditions that have a non-zero probability of occurring.