GRE Probability: Concepts, Formulas, Tips, and Rules

The GRE Quantitative Reasoning section includes a variety of challenging questions, and sometimes those challenges involve dice rolls, card shuffles, and the ever-present coin flip. Don’t let probability problems throw your GRE score off track! While they may seem difficult at first glance, these questions can actually be an opportunity to show your critical thinking and analytical skills.

In this blog, we’ll provide you with the knowledge and strategies to confidently solve these problems and add valuable points to reach your target GRE score. We’ll break down the core concepts you need to understand, explore different question formats, and provide you with valuable test-taking tactics to approach even the trickiest probability problems with ease.

What is GRE Probability?

The GRE probability uses numbers to express the likelihood of events in experiments. The GRE Quant section tests four main areas of math: arithmetic, algebra, geometry, and data analysis (which includes probability). While probability is less emphasised compared to the others, understanding it can still benefit your score.

The GRE probability questions can appear in multiple-choice, comparison, or numeric entry formats. The answer format (fractions or decimals) is usually indicated by the question itself or the answer choices. For numeric entry questions, a single blank means your answer can be a decimal or integer, while a double blank (like a fraction) requires a fraction. Don’t worry about simplifying fractions, as any equivalent form is considered correct on the GRE.

Key Concepts to Master the GRE Probability

1. Understand basic probability concepts

Probability helps us understand the likelihood of events happening. Before we start, let’s build a strong foundation with some key concepts:

• Sample space: Imagine a box containing all the possible outcomes of an experiment, like flipping a coin. The sample space for the coin flip would include “heads” and “tails.”
• Outcomes: These are the individual results within the sample space. In our coin toss example, “heads” and “tails” are the two possible outcomes.
• Events: Now, let’s focus on specific groups of outcomes. An event could be “getting heads” or “getting tails.” Events are essentially subsets of the entire sample space.
• Favourable outcomes: Not all outcomes are created equal! When we’re interested in a specific success, like getting heads, those particular outcomes are called “favourable” for that event.

2. Learn probability formulas

Here are some fundamental ones you’ll encounter often on the GRE exam. These are just a few basic formulas, and there are many more depending on the specific scenario you’re dealing with. For instance, if you’re working with multiple dice rolls or card shuffles, you might need formulas for permutations or combinations.

Basic Probability

This is the most essential formula: P(E) = Favourable Outcomes / Total Possible Outcomes. 

Here, P(E) represents the probability of event E. You calculate it by dividing the number of outcomes that satisfy your condition (favourable outcomes) by the total number of potential outcomes (sample space).

Complement Rule

This tells you the probability of an event NOT happening. P(not E) = 1 - P(E). P(not E) signifies the probability of the complement of event E, which is simply 1 minus the probability of E itself.

Conditional Probability

This formula helps you determine the probability of event B happening, given that event A already occurred. P(B|A) = P(A and B) / P(A).

P(B|A) represents the conditional probability of B given A. You calculate it by dividing the probability of both A and B happening (joint probability) by the probability of just A happening.

Independent Events

Events are considered independent if the occurrence of one doesn’t affect the probability of the other. If events A and B are independent, then P(A and B) = P(A) * P(B). This formula calculates the probability of both independent events A and B happening. You simply multiply the probabilities of each individual event.

3. Probability distributions

Imagine you’re flipping a coin. There are two possible outcomes: heads or tails. A probability formula can tell you the chance of getting heads (which is 1/2). But what if you’re flipping the coin multiple times?

A probability distribution comes into play when you have a random variable, like the number of heads in multiple coin flips. It describes the likelihood of each possible value (number of heads) that this variable can take.

There are two main types of probability distributions:

• Discrete Probability Distribution: This applies to situations where the random variable can take only certain separated values. Rolling a dice is a good example. The possible values are 1, 2, 3, 4, 5, and 6. The distribution shows the probability of each of these values occurring.
• Continuous Probability Distribution: This applies to situations where the random variable can take any value within a specific range. For instance, measuring the height of people. The heights can range from very short to very tall, and the distribution shows the probability of finding someone with a particular height within that range.

There are many different probability distributions, each with its own characteristics and applications. Some common ones include:

• Normal Distribution (Bell Curve): This symmetrical bell-shaped curve describes many natural phenomena like heights, weights, or test scores.
• Binomial Distribution: This applies to experiments with a fixed number of trials and only two possible outcomes (success/failure). Flipping a coin 10 times is a good example.
• Poisson Distribution: This describes the probability of a certain number of events happening in a fixed interval of time or space. For example, the number of customer arrivals in a store in a given hour.

When dealing with probability distributions on the GRE, your success hinges on understanding the nature of your random variable and the type of experiment involved. This knowledge empowers you to choose the correct distribution, which then streamlines your approach and enhances your efficiency.

4. Conditional probability

Conditional probability is a crucial concept for the GRE quant section, especially when dealing with probability word problems. It essentially helps you understand the likelihood of one event happening, considering another event has already occurred.

Imagine you’re flipping a fair coin. Normally, the probability of getting heads (or tails) is 1/2. But what if you’re told someone already flipped the coin and it landed on heads (Event A), and they DIDN’T show you the result (so it could be heads or tails)? Now, what’s the probability of getting tails (Event B) on your flip?

In this scenario, knowing the first coin landed heads (Event A) changes the situation for your flip (Event B). That’s because you’re not considering all possible outcomes (both heads and tails from the first flip)—you’re only considering the possibility that the first flip was heads.

Here’s how to calculate conditional probability using a formula:

P(B|A) = P(A and B) / P(A)

• P(B|A) represents the probability of event B happening given that event A already happened.
• P(A and B) represents the probability of both event A and event B happening.
• P(A) represents the probability of event A happening.

Back to the coin example:

P(tails given heads) = P(heads and tails) / P(heads)

Since the flips are independent (the outcome of the first flip doesn’t affect the second), P(heads and tails) is simply the probability of getting tails (1/2). P(heads) is also 1/2 (given the scenario). So, the answer is:

P(tails given heads) = (1/2) / (1/2) = 1

This means that even if you know the first flip was heads, the probability of getting tails on your flip remains unchanged at 1/2.

Here’s another GRE-style example:

You have a bag with 4 red marbles and 6 blue marbles. You pick a marble, don’t put it back, and then pick another. What’s the probability of picking a red marble second, given that you picked a red marble first?

P(picking a red marble second given picking red first) = P(picking two red marbles) / P(picking a red marble)

Here, P(picking two red marbles) = (3/9 * 2/8) since after taking out the first red marble, there are only 3 red remaining, and 8 total. P(picking a red marble) = 4/10. So:

P(picking a red marble second given picking red first) = (3/36) / (4/10) = 5/12

Understanding conditional probability will help you solve these kinds of GRE problems where the likelihood of an event changes based on the information you have about a previous event.

Suggested: 15 Hardest GRE Math Questions with Answers

What are the Probability Rules?

Mastering probability rules is essential for your GRE quantitative reasoning section. These rules provide a guide for solving even the most challenging probability questions by dictating how probabilities interact and behave.

The Addition Rule

This rule comes into play when you need to find the probability of either of two events happening. It states that P(A ∪ B) = P(A) + P(B) – P(A ∩ B). The tricky part here is the overlap. P(A ∩ B) represents the possibility that both events A and B occur simultaneously, and we need to subtract this overlap to avoid counting it twice.

The Multiplication Rule

This rule helps you determine the probability of two independent events happening together. It states that P(A ∩ B) = P(A) * P(B), but only if events A and B are independent (meaning the outcome of one doesn’t affect the other). If they are dependent, we use conditional probability: P(A ∩ B) = P(A) * P(B|A), where P(B|A) is the probability of B happening given that A already happened.

The Complement Rule

This rule is a shortcut for situations where calculating the probability of an event not happening (the complement) is easier. Mathematically, P(A') = 1 – P(A), where A' represents the complement of event A. Essentially, you calculate the probability of everything else happening (which is 1) and subtract the probability of event A itself.

The Total Probability Rule

This rule is for scenarios where your desired event can occur in multiple, mutually exclusive ways. It states that P(A) = Σ P(A ∩ Bi), where Bi represents all the different exclusive ways event A can occur. We basically add up the probabilities of event A happening with each of these exclusive possibilities (Bi).

8 Tips to Ace the GRE Probability

Here’s a breakdown of effective strategies to master probability on the GRE:

1. Think and visualise

Don’t just read the problem. Draw diagrams, create tables, or build charts to understand the situation clearly. For instance, use tree diagrams to map pathways and outcomes in problems with multiple probability events.

2. Simplify the problem

Complex problems can be overwhelming. Simplify them by dividing them into smaller, manageable parts. Solve each part individually, then combine the solutions to reach the final answer. Imagine a multi-step question with dice rolls. Solve the probability of each roll one by one.

3. Real-world practice

Bridge the gap between theory and application. Apply these concepts to real-life situations. Think about calculating the probability of drawing a specific card during a card game—a relatable example that clarifies abstract concepts.

4. Use Venn diagrams

These diagrams excel at visualising relationships between sets and calculating overlapping probabilities. They're particularly helpful when dealing with problems like selecting coloured balls from different bags.

5. Combinations and permutations

Master combinations and permutation concepts as they frequently appear in probability questions. Understand the difference between them and practice solving problems that involve arranging or selecting objects.

6. Context

Analyse the context of the events you’re dealing with. Are they mutually exclusive (can’t happen together), independent (unaffected by each other), or dependent (one outcome influences the other)? This understanding helps you choose the appropriate probability formula for accurate calculations.

7. Don’t miss the hidden details

Read the question meticulously. Sometimes, there are some hints or restrictions that impact your answer. For example, consider a question asking about drawing two cards from a deck without replacement. This implies the deck composition changes after the first draw, affecting the probability of the second draw.

8. Practice makes perfect

As with any skill, consistent practice is key. Get your hands on GRE practice books and materials, focusing specifically on probability questions. Regular practice hones your problem-solving skills, builds confidence, and familiarises you with the diverse range of scenarios you might come across on the GRE.

Suggested: 21 Most Difficult GRE Math Formulas for 2024

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From the Desk of Yocket

Probability questions on the GRE tend not to be a major focus of the exam. Typically, only a few questions are there per test. This means you can potentially avoid a significant portion of the quant section if you’re not a pro at statistics. However, probability questions can also be a strategic advantage.

They often test logic and critical thinking skills more than complex calculations. This means that extensive memorization of complex formulas isn’t necessary. But on the other hand, these questions can test your critical thinking and analytical skills uniquely. If you have a strong grasp of the core concepts, you can approach these problems with a clear understanding and potentially outperform others who might struggle with unfamiliar territory. Furthermore, probability questions can sometimes be solved more quickly than other quant problems because they often involve smaller datasets and more intuitive reasoning. Prepare effectively for GRE probablitiy with Yocket GRE prep to enhance your critical thinking skills and tackle the exam confidently.