Mastering
Probability
Build a rock-solid foundation in probability: from basic definitions to compound events, with strategies tailored for the GRE.
What Is Probability?
Probability measures how likely an event is to occur. It is always a number between \(0\) and \(1\), inclusive. An event that cannot happen has probability \(0\); an event that is certain has probability \(1\).
For example, the probability of rolling a \(3\) on a fair six-sided die is \(\displaystyle P(3) = \frac{1}{6}\).
Sample Space and Events
The Probability Scale
Every probability value lives on this scale. On the GRE, if your answer falls outside \([0, 1]\), something went wrong.
The Addition Rule
We subtract \(P(A \cap B)\) because the overlap is counted twice when we add \(P(A)\) and \(P(B)\).
Mutually Exclusive Events
Two events are mutually exclusive (disjoint) if they cannot both happen at the same time. Their intersection is empty: \(A \cap B = \emptyset\).
Independent vs Dependent
The Multiplication Rule
Use multiplication when you need both events to occur ("A and B"). Use addition when you need at least one to occur ("A or B").
With vs Without Replacement
| With Replacement | Without Replacement | |
|---|---|---|
| Independence | Independent | Dependent |
| Sample size | Stays the same | Decreases by 1 each draw |
| Formula | \(P(A) \cdot P(B)\) | \(P(A) \cdot P(B \mid A)\) |
| Example | Rolling a die twice | Drawing 2 cards from a deck |
How to Solve Probability Problems
Example 1: Single Event
A jar contains 5 blue, 3 red, and 2 green marbles. If one marble is drawn at random, what is the probability it is red?
Example 2: Two Events (And)
A fair coin is flipped and a fair six-sided die is rolled. What is the probability of getting heads and rolling a number greater than 4?
Example 3: Without Replacement
A bag has 4 blue and 6 red balls. Two balls are drawn without replacement. What is the probability both are red?
The Complement Strategy
When a problem says "at least one," it is almost always faster to compute the complement: the probability that none occur, then subtract from \(1\).
Common Mistakes
Probability Formula Cheat Sheet
| Scenario | Formula | Key Word |
|---|---|---|
| Basic probability | \(\frac{\text{favorable}}{\text{total}}\) | — |
| A or B | \(P(A)+P(B)-P(A \cap B)\) | OR |
| A or B (mutually exclusive) | \(P(A) + P(B)\) | OR (no overlap) |
| A and B (independent) | \(P(A) \cdot P(B)\) | AND |
| A and B (dependent) | \(P(A) \cdot P(B \mid A)\) | AND (conditional) |
| Complement | \(1 - P(E)\) | NOT / at least one |
Question 1 of 5
A bag contains 8 red and 12 blue marbles. One marble is drawn at random. What is \(P(\text{blue})\)?
Question 2 of 5
Two fair dice are rolled. What is the probability that both show a number less than 3?
Question 3 of 5
Events \(A\) and \(B\) are mutually exclusive with \(P(A) = 0.3\) and \(P(B) = 0.5\). What is \(P(A \cup B)\)?
Question 4 of 5
From a standard 52-card deck, two cards are drawn without replacement. What is the probability both are aces?
Question 5 of 5
A fair coin is flipped 4 times. What is the probability of getting at least one tail?