Probability
Fundamentals
Master the core rules of probability: from basic definitions to compound events and conditional reasoning for the SAT.
What Is Probability?
Probability measures how likely an event is to occur. It is always a number between \(0\) and \(1\), inclusive. A probability of \(0\) means impossible; a probability of \(1\) means certain.
On the SAT, you will use this formula with tables, surveys, and descriptions of experiments. The key is counting outcomes correctly.
The Probability Scale
Key Vocabulary
The Addition Rule
The Multiplication Rule
When two events are independent (the outcome of one does not affect the other), you multiply their probabilities to find the probability of both occurring.
Independent vs. Dependent
| Independent | Dependent | |
|---|---|---|
| Definition | Outcome of A does not affect B | Outcome of A changes probability of B |
| Test | \( P(B \mid A) = P(B) \) | \( P(B \mid A) \neq P(B) \) |
| Formula | \( P(A) \times P(B) \) | \( P(A) \times P(B \mid A) \) |
| Example | Flip coin, then roll die | Draw 2 cards without replacement |
Conditional Probability
Conditional probability answers: "Given that one event has happened, what is the probability of another?" This appears frequently in SAT two-way table questions.
How to Solve SAT Probability
Example 1: Basic Probability
A bag contains 5 blue, 3 red, and 2 green marbles. If one marble is chosen at random, what is the probability it is red?
Example 2: Two-Way Table
The table below shows survey results for 200 students. What is the probability that a randomly selected student prefers basketball, given that the student is a junior?
Example 3: Compound Events
A standard deck has 52 cards. What is the probability of drawing a heart or a king?
Key takeaway: Always subtract the overlap when events can happen simultaneously.
Example 4: Using the Complement
A jar has 20 gumballs. 4 are yellow. If one gumball is chosen at random, what is the probability it is not yellow?
Both methods give the same answer. The complement approach is especially useful when counting "not" directly is harder.
Common Mistakes
Probability Formula Summary
| Rule | Formula | When to Use |
|---|---|---|
| Basic | \( \dfrac{\text{favorable}}{\text{total}} \) | Single event, no conditions |
| Complement | \( 1 - P(A) \) | "Not A" or "at least one" |
| Addition | \( P(A) + P(B) - P(A \text{ and } B) \) | "A or B" |
| Multiplication (ind.) | \( P(A) \times P(B) \) | "A and B," independent |
| Conditional | \( \dfrac{P(A \text{ and } B)}{P(A)} \) | "B given A" |
Question 1 of 5
A box contains 8 red and 12 blue balls. One ball is drawn at random. What is \( P(\text{blue}) \)?
Question 2 of 5
A fair coin is flipped \(3\) times. What is the probability of getting heads on all \(3\) flips?
Question 3 of 5
In a class of \(30\) students, \(18\) play soccer and \(10\) play tennis. \(5\) play both. What is \( P(\text{soccer or tennis}) \)?
Question 4 of 5
A survey asked \(150\) employees whether they commute by car or train.
Results: \(60\) drive a car, \(70\) take the train, and \(20\) use both. The rest use neither.
Given that an employee drives a car, what is the probability they also take the train?
Question 5 of 5
The probability that it rains on Saturday is \( 0.4 \). The probability that it rains on Sunday is \( 0.3 \). If these events are independent, what is the probability it rains on neither day?