Introduction
Conditional probability questions are a reliable part of the Problem-Solving and Data Analysis domain (roughly 15% of the SAT Math section). They test one specific skill: when the problem tells you a condition has already occurred, you must restrict your universe to only the people or outcomes that satisfy that condition — and that restriction changes the denominator. Two-way tables are the most common data display for this skill, so reading them precisely is non-negotiable.
Scope of this note: Reading two-way frequency tables, choosing the correct denominator (row total, column total, or grand total), and interpreting "given that" language. For straightforward single-event probability with one-variable displays, see the sibling note Probability from Data.
Core Concept
A two-way frequency table cross-classifies a group of people (or items) by two categorical variables. Each cell holds a count. The margins — the row totals, column totals, and grand total — are where denominator decisions happen.
Three types of probability from a two-way table:
| Question type | Denominator | Example phrasing |
|---|---|---|
| Simple (unconditional) | Grand total | "What is the probability that a randomly selected person…?" |
| Conditional on a row category | That row total | "Given that the person is in Group A, what is the probability…?" |
| Conditional on a column category | That column total | "Given that the person chose Option X, what is the probability…?" |
The keyword "given that" (or equivalents: "knowing that," "among those who," "of the students who") tells you a condition is active. The condition names a row or column, and you must use only that row or column as your new sample space.
Quick illustration:
Suppose a table shows 200 students:
| Plays instrument | Does NOT play instrument | Row total | |
|---|---|---|---|
| Plays sport | 54 | 66 | 120 |
| Does NOT play sport | 22 | 58 | 80 |
| Column total | 76 | 124 | 200 |
- — grand total denominator, no condition.
- — the condition "plays sport" locks you into the top row; that row's total (120) is the new denominator.
- — the condition "plays instrument" locks you into the first column; that column's total (76) is the new denominator.
Notice that (the overlap cell) is always the numerator — the joint count of both events. Only the denominator changes with the condition.
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