Compute the probability of single events, complements, and compound events using the addition and multiplication rules, including independent and mutually exclusive events (Statistics and Probability).

What this topic is asking

Probability measures how likely an event is, as a number from 0 (impossible) to 1 (certain). The ACT tests the basic ratio of favorable to total outcomes, the complement rule, and compound events using the multiplication rule (independent events) and addition rule (mutually exclusive events).

Basic probability

Every probability starts from counting outcomes.

The complement rule is a powerful shortcut: when "at least one" is asked, it is often easier to compute the probability of none and subtract from 1.

Compound events: "and" versus "or"

The two compound rules are the heart of ACT probability.

Read the wording: "and" (both happen) usually means multiply; "or" (either happens) usually means add.

Using the complement for "at least one"

When a question asks for "at least one" success, the complement is "none", which is often a single multiplication. For example, the probability of at least one head in three coin flips is $1 - P(\text{no heads}) = 1 - \left(\frac{1}{2}\right)^{3} = 1 - \frac{1}{8} = \frac{7}{8}$. Computing the "none" case and subtracting from 1 is much faster than adding up the probabilities of exactly one, two and three heads.

Dependent events and drawing without replacement

When outcomes do affect each other, the probabilities change between steps. Drawing two marbles without replacement is dependent: if a bag has 5 marbles with 2 red, the chance both draws are red is $\frac{2}{5}\times \frac{1}{4}$, because after removing one red, only 1 red remains among 4 marbles. Adjusting the second probability for the changed contents is the key skill here. With replacement, the draws are independent and the probability stays the same each time.

Why reading the wording matters

Most probability errors come from misreading "and" versus "or", or from forgetting that drawing without replacement changes the second probability. The reliable approach is to identify whether events are independent (multiply), mutually exclusive (add), or dependent (adjust the later probabilities), and to use the complement whenever "at least one" appears. Reducing the final fraction to lowest terms matches the answer choices the ACT provides.

Try this

Q1. A die is rolled. What is the probability of rolling a number greater than 4? [1 point]

• Cue. Favorable outcomes are 5 and 6, so $\frac{2}{6} = \frac{1}{3}$.

Q2. Two independent events have probabilities $\frac{1}{2}$ and $\frac{1}{5}$. What is the probability both occur? [1 point]

• Cue. Multiply: $\frac{1}{2}\times \frac{1}{5} = \frac{1}{10}$.