GRE Conditional Probability Questions Practice Questions with Answers

Concept Explanation

Conditional probability is the likelihood of an event occurring given that another event has already taken place. In the context of the Graduate Record Examination, this concept measures how the sample space of a problem is restricted by a specific piece of information. The standard formula for the probability of event $A$ occurring given that event $B$ has occurred is denoted as $P(A|B)$ and is calculated using the formula $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$ where $P(A \cap B)$ represents the probability that both events happen and $P(B)$ is the probability of the condition itself. This topic is a frequent component of the GRE Prep curriculum, specifically within the Data Analysis section.

When solving GRE conditional probability questions, the most effective strategy is often to redefine the denominator. Instead of looking at the entire original population, you focus only on the subset that meets the given condition. For example, if you are asked for the probability that a student is a math major given they are a senior, your "total" becomes only the seniors, not the whole student body. Understanding the difference between independent events—where the occurrence of one does not affect the other—and dependent events is crucial for accuracy. You can find more structured practice for quantitative reasoning using the AI Question Generator to target these specific logic gaps.

Solved Examples

Review these worked examples to understand how to apply the conditional probability formula in different GRE-style scenarios.

1. Example 1: The Marble Bag
   A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn without replacement. What is the probability that the second marble is blue, given that the first marble drawn was red?
   1. Identify the condition: The first marble drawn was red.
   2. Determine the new sample space: After removing one red marble, there are 4 red and 3 blue marbles left (7 total).
   3. Calculate the probability: The number of blue marbles (3) divided by the remaining total (7).
   4. Solution: $P( \text{Blue}_2| \text{Red}_1) = \frac{3}{7}$.
2. Example 2: Two-Way Tables
   In a group of 100 people, 40 are male and 60 are female. Of the males, 10 are smokers. Of the females, 20 are smokers. If a smoker is picked at random, what is the probability that the person is female?
   1. Identify the condition: The person is a smoker.
   2. Calculate the total number of smokers: $10 \text{ (males)} + 20 \text{ (females)} = 30$.
   3. Identify the target group within the condition: Female smokers (20).
   4. Apply the formula: $\frac{20}{30} = \frac{2}{3}$.
3. Example 3: Independent Events
   If $P(A) = 0.5$ and $P(B) = 0.2$, and events $A$ and $B$ are independent, what is $P(A|B)$?
   1. Recall the definition of independence: If events are independent, the occurrence of $B$ does not change the probability of $A$.
   2. Apply the rule: $P(A|B) = P(A)$.
   3. Solution: $P(A|B) = 0.5$.

Practice Questions

Test your skills with these GRE conditional probability questions. Ensure you read the "given that" statements carefully.

1. A fair six-sided die is rolled. What is the probability that the result is a 6, given that the result is an even number?
2. A box contains 4 green balls and 6 yellow balls. Two balls are drawn one after another without replacement. What is the probability that both are green?
3. In a survey of 50 students, 30 like chocolate, 20 like vanilla, and 10 like both. If a student who likes chocolate is chosen at random, what is the probability they also like vanilla?

4. Two cards are drawn from a standard deck of 52 cards without replacement. What is the probability that the second card is a Heart, given that the first card was a Heart?
5. If $P(X) = 0.6$, $P(Y) = 0.4$, and $P(X \cap Y) = 0.12$, are events $X$ and $Y$ independent? Calculate $P(X|Y)$.
6. A weather forecast states there is a 70% chance of rain. If it rains, there is a 40% chance of heavy wind. What is the probability that it rains and there is heavy wind?
7. In a class, 60% of students passed Math, 70% passed English, and 50% passed both. What is the probability that a student passed English, given that they passed Math?
8. A drawer contains 10 black socks and 10 white socks. If you pull out two socks at random, what is the probability the second is black given the first one was white?
9. A jar has 3 red, 2 white, and 5 blue marbles. If a marble is drawn and it is known not to be blue, what is the probability it is red?
10. If $P(A) = 0.3$, $P(B) = 0.4$, and $P(B|A) = 0.5$, find $P(A \cap B)$.

Answers & Explanations

1. 1/3: The even numbers on a die are {2, 4, 6}. Out of these 3 possibilities, only one is a 6. Thus, $1/3$.
2. 2/15: The probability the first is green is $4/10$. Given the first is green, the probability the second is green is $3/9$. Multiply them: $(4/10) \times (3/9) = 12/90 = 2/15$.
3. 1/3: There are 30 students who like chocolate (the condition). Of those 30, 10 also like vanilla. $10/30 = 1/3$.
4. 12/51 or 4/17: A deck has 13 Hearts. If one is removed, 12 Hearts remain out of 51 total cards.
5. Independent; 0.6: Check independence: $0.6 \times 0.4 = 0.24$. Since $0.24 \neq 0.12$, they are NOT independent. However, $P(X|Y) = P(X \cap Y) / P(Y) = 0.12 / 0.4 = 0.3$. (Correction: They are dependent).
6. 0.28 or 28%: Using the multiplication rule: $P(Rain \cap Wind) = P(Rain) \times P(Wind|Rain) = 0.70 \times 0.40 = 0.28$.
7. 5/6: $P(English|Math) = P(Math \cap English) / P(Math) = 0.50 / 0.60 = 5/6$.
8. 10/19: After picking one white sock, 19 socks remain, 10 of which are black.
9. 3/5: If it is not blue, the marble must be red (3) or white (2). Total sample space is $3+2=5$. Red is $3/5$.
10. 0.15: Using $P(B|A) = P(A \cap B) / P(A)$, we get $0.5 = P(A \cap B) / 0.3$. Multiplying gives $0.15$.

1. If two events A and B are mutually exclusive, what is P(A|B)?

• A0
• B1
• C0.5
• DP(A)

Frequently Asked Questions

What is the difference between P(A and B) and P(A|B)?

P(A and B) is the probability that both events occur simultaneously out of the entire sample space. P(A|B) is the probability that A occurs specifically within the subset where B has already happened.

Does "without replacement" always imply conditional probability?

Yes, because removing an item changes the total count and the composition of the remaining items. This means the second event's probability is dependent on the outcome of the first event.

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

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

Can conditional probability be greater than 1?

No, like all probabilities, conditional probability must be between 0 and 1 inclusive. It is a ratio of the intersection of two events to the probability of the condition.

What if P(B) is zero in the conditional probability formula?

If the probability of the condition P(B) is zero, the conditional probability P(A|B) is undefined. This is because you cannot condition an event on something that is impossible to occur.