AP Statistics · Topic 2.4 · Unit 2
AP Stats 2.4: Introduction to Probability
By Jude Wallis · Published
Topic 2.4 in the Fall 2026 AP Statistics course defines probability. The sample space of all outcomes has probability 1, every probability lies between 0 and 1, an equally likely event equals favorable over total outcomes, and the complement of E has probability 1 minus P(E).
AP Statistics: Unit 2 (topics 2.4). In the Fall 2026 AP Statistics course, introduction to probability is Unit 2 Topic 2.4, aligned to skill 3.C.
What topic 2.4 covers
Topic 2.4 gives probability its formal definitions after the simulation warm-up in topic 2.3. The objective is to calculate probabilities for events and their complements.
The sample space of a random process is the set of all possible nonoverlapping outcomes, and the probability of the whole sample space is 1. Every event is a collection of outcomes drawn from that sample space.
Theoretical probability and the 0 to 1 scale
When all outcomes in the sample space are equally likely, the theoretical probability that an event occurs is the number of outcomes in divided by the total number of outcomes in the sample space. This probability is written .
Every probability is a number between 0 and 1, inclusive. A probability of 0 means the event never happens, a probability of 1 means it always happens, and values in between measure how likely the event is. If a calculation ever lands outside that range, something has gone wrong.
The complement rule
The complement of an event is the event that does not happen, written , , or and read as 'not E'. Because either happens or it does not, the two probabilities add to 1.
That gives the complement rule: . It is a small tool with large payoff, because 'at least one' and 'none' questions are often far easier to answer through the complement than head-on. You will reuse it constantly, from the complement of independent events to binomial 'at least' problems.
Where these definitions lead
These few rules are the foundation for the rest of Unit 2. Mutually exclusive events, conditional probability, and the addition and multiplication rules all build on the sample space and the 0 to 1 scale you set up here.
Unit 2 makes up 15 to 25 percent of the multiple-choice section, and probability rules run through much of it, so getting these basics exact pays off repeatedly. A quick self-check on any probability answer is whether it lands between 0 and 1: a negative value or a value above 1 signals a setup error, most often a miscounted sample space. Equally likely counting also has limits, since it only applies when every outcome truly has the same chance.
Even rolls and a complement on a fair die
Roll a fair six-sided die once. The sample space is {1, 2, 3, 4, 5, 6}, six equally likely outcomes. Find the probability of an even roll, and the probability of not rolling a 5.
Even outcomes are {2, 4, 6}, so .
Rolling a 5 is a single outcome, so .
Apply the complement rule for not a 5: .
Compute: .
and .
Frequently asked questions
What is a sample space in probability?
The sample space of a random process is the set of all possible nonoverlapping outcomes. For one roll of a die it is {1, 2, 3, 4, 5, 6}. The probability of the entire sample space is always 1, because some outcome in it is certain to happen.
What is the complement rule?
The complement of an event E is 'not E', written E prime, E-bar, or E to the C. The complement rule says P(not E) equals 1 minus P(E). It is useful because 'at least one' questions are often easier to solve by finding the probability of 'none' first and subtracting from 1.