Digital SAT · Probability & Two-Way Tables
How to Solve Probability from a Two-Way Table on the Digital SAT
A two-way table sorts a group by two categories at once. Probability is still favorable over total, but which total you use depends on the question. A plain probability divides the matching cell by the grand total. A conditional probability, flagged by given that, restricts the total to just that row or column. Reading the correct denominator is the whole skill.
- per test
- less than 1 per test per test
- typical difficulty
- Mostly easy typical difficulty
- practice questions
- 45 practice questions
Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.
What the question bank shows
Plain uses the grand total; conditional uses the row or column after given that.
A table-reading task; the challenge is the right denominator.
The phrase given that shrinks the total to a single row or column.
How to recognize probability from a two-way table questions
- A table cross-classifies a group by two categories, with row and column totals.
- The question asks for the probability of a category, or a conditional one using given that.
- The words probability, given that, among, or of those appear.
- Choices are fractions, decimals, or percents.
Why students miss these
The step-by-step method
- 1
Find the favorable cell
Locate the cell or cells that match the event in the numerator.
- 2
Decide the total
A plain probability uses the grand total; a conditional one uses the row or column named after given that.
- 3
Read the right denominator
Pull the grand total, row total, or column total the question requires.
- 4
Divide and match
Divide favorable by total, then convert to the fraction, decimal, or percent the choices use.
Worked examples
Easy example
| Genre | Number of books |
|---|---|
| Fiction | 20 |
| Mystery | 10 |
| History | 15 |
| Poetry | 5 |
A: The probability is the number of favorable outcomes over the total: .
B: This is the odds (favorable to unfavorable), , not the probability.
C: This miscounts the favorable outcomes as 2 instead of 1.
D: This is the complement — the probability of NOT selecting the target, .
Explanation
Mystery books (10) over the total (50): .
Detailed explanation
The total is , and 10 are mystery, so the probability is .
Medium example
| Grade | Buys | Brings | Total |
|---|---|---|---|
| Grade 9 | 30 | 30 | 60 |
| Grade 10 | 45 | 15 | 60 |
| Total | 75 | 45 | 120 |
A: The probability is the number of favorable outcomes over the total: .
B: This miscounts the favorable outcomes as 4 instead of 3.
C: This is the odds (favorable to unfavorable), , not the probability.
D: This is the complement — the probability of NOT selecting the target, .
Explanation
Grade 10 and buys (45) over all students (120): .
Detailed explanation
Across all 120 students, 45 are both in grade 10 and buy lunch, so the probability is .
Hard example
| Age group | Online | In person | Total |
|---|---|---|---|
| Under 18 | 20 | 10 | 30 |
| 18 to 39 | 60 | 30 | 90 |
| 40 or older | 30 | 50 | 80 |
| Total | 110 | 90 | 200 |
A: Incorrect. This divides by the grand total of 200 instead of the number who are 40 or older.
B: Correct. Among the 80 attendees who are 40 or older, 30 purchased online, so the probability is .
C: Incorrect. This uses the in-person column total (90) as the denominator instead of the 40-or-older total.
D: Incorrect. This is the probability that an attendee 40 or older purchased in person, the complement of what is asked.
Explanation
Restrict to attendees 40 or older (80); 30 bought online: .
Detailed explanation
Because the condition is the 40-or-older age group, the denominator must be those 80 attendees, and 30 of them purchased online, giving .
The common traps
| Pattern | What it does | The tell |
|---|---|---|
| Wrong total for a conditional | Used the grand total when the question restricted to a row or column. | Given that names the group; its total is the denominator. |
| Wrong cell | Read a neighboring cell for the numerator. | Match the favorable cell to both categories in the event. |
| Row versus column | Used the row total when the column applied, or the reverse. | Identify whether the condition fixes a row or a column. |
Try it: two real questions
| Activity | Number of members |
|---|---|
| Art | 16 |
| Music | 8 |
| Drama | 12 |
| Dance | 4 |
The table shows the number of club members who chose each activity. If one member is selected at random, what is the probability that the selected member chose art?
| Pet | Number of students |
|---|---|
| Dog | 18 |
| Cat | 12 |
| Fish | 6 |
| None | 4 |
The table shows the number of students in a class who own each type of pet. If one student is selected at random, what is the probability that the selected student owns no pet?
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Related reading
Common questions
How do you find probability from a two-way table on the SAT?
Find the cell that matches the event for the numerator, then choose the total: the grand total for a plain probability, or a row or column total for a conditional one. Divide favorable by total.
What is a conditional probability in a table?
A probability restricted to a subgroup, signaled by given that or among. The denominator becomes that group's total, a single row or column, not the grand total.
How do I know which total to use?
If the question limits the group, as in given that a person is a senior, use that row or column total. If it asks about the whole group, use the grand total in the corner.
Do I need a calculator for two-way table probability?
Only to divide. The thinking is reading the table: picking the correct cell and total. Once those are set, the division is quick.
Practice probability from a two-way table the way it is tested
Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.
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