GMAT Probability Cheat Sheet

Preparing for the GMAT adaptive test can be challenging, particularly when it comes to mastering probability. This GMAT Probability Cheat Sheet aims to clarify the essential concepts and formulas necessary for effectively addressing probability questions. By following this guide, you will gain a comprehensive understanding of how to approach and solve probability problems on the GMAT.

Understanding GMAT Probability

Probability is a key topic tested on the GMAT quantitative question section. It deals with measuring the likelihood or chance of an event occurring. Understanding probability concepts and formulas is crucial for solving GMAT probability questions effectively.

Probability Formula

The fundamental probability formula is:

P(A) = Number of favourable outcomes / Total number of possible outcomes. 

This formula gives the probability of event A occurring. The numerator is the count of outcomes where the event occurs, and the denominator is the total count of all possible outcomes.

Types of Probability Questions on GMAT

The GMAT quantitative section often tests probability concepts through a variety of question types. Here are some of the common types of probability questions you may encounter:

Simple Probability 

The basic probability formula is: P(A) = Number of favourable outcomes / Total number of possible outcomes.

This formula gives the probability of event A occurring. The numerator is the count of outcomes where the event occurs, and the denominator is the total count of all possible outcomes.

Example:
What is the probability of rolling a 3 on a 6-sided die?

In this case, the event A is rolling a 3 on the die. 

The number of favourable outcomes (rolling a 3) is 1.
The total number of possible outcomes (rolling 1, 2, 3, 4, 5, or 6) is 6.

Plugging these values into the probability formula: P(rolling a 3) = 1 / 6

Therefore, the probability of rolling a 3 on a 6-sided die is 1/6 or approximately 0.167 or 16.7%.

Compound Probability 

Compound probability refers to the probability of two or more events occurring together. This is used when the problem statement asks for the likelihood of the occurrence of more than one outcome.

The formula for compound probability is: P(A and B) = P(A) * P(B)

Where P(A and B) is the probability of both events A and B occurring.

This formula applies when the events A and B are independent, meaning the occurrence of one event does not affect the probability of the other event.

Example: A jar contains 4 red balls and 6 blue balls. What is the probability of randomly selecting a red ball and then a blue ball, without replacement?

To solve this:

• There are 10 balls total in the jar (4 red, 6 blue)
• The probability of selecting a red ball on the first draw is 4/10 = 2/5
• After selecting the red ball, there are now 9 balls remaining (3 red, 6 blue)
• The probability of selecting a blue ball on the second draw is 6/9

Using the compound probability formula: P(red and blue) = P(red) * P(blue)
= 2/5 * 6/9
= 12/45
= 4/15

Therefore, the probability of randomly selecting a red ball and then a blue ball, without replacement, is 4/15.

Mutually Exclusive Events 

Mutually exclusive events are events that cannot occur simultaneously. In other words, if one event occurs, the other event cannot occur at the same time. The key properties of mutually exclusive events are:

• The probability of both events occurring is zero: P(A and B) = 0
• The probability of either event occurring is the sum of their individual probabilities: P(A or B) = P(A) + P(B)

Example:
If a coin is flipped, the outcomes of getting a head (H) or a tail (T) are mutually exclusive. If the coin lands on heads, it cannot land on tails at the same time, and vice versa. Let’s calculate the probability of getting a head or a tail on a single coin flip:

• There are 2 possible outcomes: head (H) or tail (T)
• The probability of getting a head is P(H) = 1/2 = 0.5
• The probability of getting a tail is P(T) = 1/2 = 0.5

Using the addition rule for mutually exclusive events: P(H or T) = P(H) + P(T)
= 1/2 + 1/2
= 1

The probability of getting either a head or a tail is 1, which makes sense since one of the two outcomes must occur.

Non-Mutually Exclusive Events 

Non-mutually exclusive events are events that can occur simultaneously or have some overlap in their outcomes. In other words, the occurrence of one event does not preclude the occurrence of the other event. The key properties of non-mutually exclusive events are:

• The probability of both events occurring is greater than zero: P(A and B) > 0
• The probability of either event occurring is the sum of their individual probabilities minus the probability of both occurring: P(A or B) = P(A) + P(B) – P(A and B)

Example:
Consider a group of 30 students in a math class, where 17 are boys and 13 are girls. Suppose 4 boys and 5 girls received an A grade on a test. The events we are interested in are:

• Event A: Selecting a girl from the class
• Event B: Selecting a student who received an A grade

These events are non-mutually exclusive because:

• There are 5 girls who received an A grade
• These 5 girls are part of both the “girl” group and the “A grade” group

Using the formula for non-mutually exclusive events: P(girl or A grade) = P(girl) + P(A grade) – P(girl and A grade)
= 13/30 + 9/30 – 5/30
= 17/30

The probability of selecting a girl or a student who received an A grade is 17/30.

Combinatorics and Probability

Some GMAT math probability questions involve combinatorics, which is the branch of mathematics that deals with counting.

The binomial coefficient formula nCk = n! / (k! * (n-k)!) is often used. These questions may ask you to calculate the number of ways to select items from a set, and then use that to find the probability.

Example : A group of 8 friends (4 men and 4 women) are going on a trip. They need to choose 2 people to drive the car. What is the probability that at least one of the drivers is a woman?

To solve this:

1. First, we need to calculate the total number of possible ways to choose 2 drivers from the group of 8 people.
   • This is a combination problem, where order does not matter.
   • The formula for combinations is: nCr = n! / (r! * (n-r)!)
   • In this case, n = 8 (total number of people) and r = 2 (number of people chosen)
   • So the total number of possible driver combinations is: 8C2 = 8! / (2! * 6!) = 28
2. Now we need to calculate the number of favourable outcomes – the number of ways to choose at least one woman driver.
   • This can happen in 3 ways:
     1. 2 women are chosen as drivers
     2. 1 woman and 1 man are chosen as drivers
   • The number of ways to choose 2 women out of 4 is 4C2 = 4! / (2! * 2!) = 6
   • The number of ways to choose 1 woman and 1 man out of 4 women and 4 men is 4C1 * 4C1 = 4 * 4 = 16
   • So the total number of favourable outcomes is 6 + 16 = 22
3. Finally, we calculate the probability:
   • Probability = Number of favourable outcomes / Total number of possible outcomes
   • Probability = 22 / 28 = 11/14

Therefore, the probability that at least one of the drivers is a woman is 11/14.

Conditional Probability

Conditional probability GMAT questions ask you to find the probability of an event occurring given that another event has already occurred.

The formula is: P(A|B) = P(A and B) / P(B)

These questions require you to carefully identify the relationship between the events and apply the conditional probability rule.                                                          

Example: In a group of 100 students, 60 are taking a math class and 40 are taking an English class. 20 students are taking both the math and English classes. What is the probability that a randomly selected student is taking the English class, given that the student is already taking the math class?

To solve this:

1. Let’s define the events:
   • Event A: Student is taking the math class
   • Event B: Student is taking the English class
2. We want to find P(B|A), the probability of Event B (taking English) given that Event A (taking math) has occurred.
3. We can use the formula for conditional probability:
   P(B|A) = P(A and B) / P(A)
4. From the information given:
   • P(A) = 60/100 = 0.6 (probability of taking math)
   • P(A and B) = 20/100 = 0.2 (probability of taking both math and English)
5. Plugging these values into the formula:
   P(B|A) = P(A and B) / P(A)
   P(B|A) = 0.2 / 0.6 = 1/3

Therefore, the probability that a randomly selected student is taking the English class, given that the student is already taking the math class, is 1/3.

GMAT Probability Practice Questions

Here are a few GMAT probability practice problems with explanations. Practice as many sample questions as possible to understand how to score a 750 on the GMAT.

Tips to solve GMAT probability questions 

The key is to practice GMAT probability questions regularly, master the core concepts, and develop an intuitive understanding of probability. With enough practice, you’ll be well on your way to acing the GMAT quantitative section. Understand the importance of GMAT test and follow the below tips to solve the GMAT probability section.

1. Understand Basic Probability Rule: Know that the probability of an event is calculated by dividing the number of favourable outcomes by the total number of possible outcomes. For independent events, the probability of two events both happening is the product of their individual probabilities.
2. Use GMAT Math Formulas: Familiarize yourself with GMAT quantitative topics and essential math formulas, including those for combinatorics and probability. Practice using the combination formula for problems involving selections.
3. Apply Probability Rules: Learn the rules you need to know, such as the addition rule for mutually exclusive events and conditional probability for dependent events.
4. Practice with Sample Questions: Utilize GMAT books and prep resources like the GMAT Club and the TTP GMAT Blog for practice questions.
5. Create a Probability Cheat Sheet: A GMAT math cheat sheet with key probability formulas and concepts can be a handy reference during your study sessions. Include common problems, such as calculating the probability of picking a specific item from a set or the probability that either event A or B occurs.
6. Review and Reflect: After completing practice questions, review your answers to understand your mistakes. Use GMAT math sample questions to identify areas where you need improvement.
7. Test Day Preparation: Develop a study plan that includes full-length practice tests and targeted review of probability problems. Approach the GMAT with confidence, knowing you have a strong grasp of the probability concepts tested on the GMAT.
8. Solve Sample Questions: Regularly solve sample probability GMAT questions to build confidence and proficiency.

By following these tips and regularly practicing, you’ll ultimately know how to ace the GMAT quantitative section.

Important Formulas for GMAT 

Here are the key probability formulas you need to know to crack the GMAT exam.

Conclusion

Mastering probability for the GMAT requires understanding the fundamental concepts and practicing regularly. Use this cheat sheet to familiarize yourself with key formulas and strategies. With consistent practice, you’ll be able to approach probability questions with confidence and improve your GMAT percentile.

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