Probability and Conditional Probability: Free SAT Math Lesson

"Probability and conditional probability" is an officially named skill in College Board's Problem-Solving and Data Analysis domain, and it's tested through everything from simple single-event probability to reading two-way frequency tables. This free, complete lesson covers basic probability, independent events and mutually exclusive events, reading two-way tables to find conditional probability, and the complement rule for "at least one" problems. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.

1. Basic Probability

The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes:

P(event) = number of favorable outcomestotal number of outcomes

Probability is always between 0 (impossible) and 1 (certain), and can be written as a fraction, decimal, or percent.

2. Independent and Mutually Exclusive Events

Two events are independent if the outcome of one doesn't affect the probability of the other. For independent events, multiply their individual probabilities to find the probability of both happening:

P(A and B) = P(A) × P(B)

Two events are mutually exclusive if they cannot both happen at the same time. For mutually exclusive events, add their individual probabilities to find the probability of either happening:

P(A or B) = P(A) + P(B)

3. Two-Way Tables and Conditional Probability

A two-way table organizes data by two categories at once and is a common way the SAT presents information for conditional probability questions. A conditional probability, "the probability of A given B," restricts your attention to only the row or column matching the given condition, then asks what fraction of that restricted group also satisfies the first condition.

	Owns a Car	Doesn't Own a Car	Total
Male	120	30	150
Female	90	60	150
Total	210	90	300

4. The Complement Rule

The complement of an event is everything that event does not include. Since an event and its complement together account for every possible outcome:

P(not A) = 1 − P(A)

This rule is especially useful for "at least one" problems, which are often much easier to solve by finding the probability of the opposite, "none," and subtracting from 1.

Common Mistakes to Avoid

• Adding probabilities for events that aren't mutually exclusive. The simple addition rule, P(A or B) = P(A) + P(B), only works when A and B can't both happen. If they can overlap, you need P(A) + P(B) − P(A and B) instead.
• Multiplying probabilities for events that aren't independent. If one event affects the probability of another, like drawing cards without replacement, you can't simply multiply the original probabilities together.
• Reading a conditional probability from the wrong row or column of a two-way table. "P(A given B)" means restrict to the B row or column first, then find what fraction of that group also satisfies A. Reversing the order gives a different, incorrect answer.
• Confusing "and" probability with conditional probability in a two-way table. P(A and B) divides the joint count by the grand total. P(A given B) divides the joint count by only B's row or column total. These use different denominators and give different answers.
• Overcomplicating an "at least one" problem instead of using the complement. Directly calculating "at least one success" often requires adding up several cases. Finding the complement, "zero successes," and subtracting from 1 is almost always faster.

Frequently Asked Questions