# Probability and two-way tables on the SAT

Canonical: https://duckyhelper.com/learn/sat-math/probability/
Updated: 2026-10-01

Probability is the number of outcomes you want divided by the total number of outcomes, so it is between 0 and 1. Most SAT probability questions use a two-way table. The whole skill is finding the right total. Words like "given that" or "of the students who" shrink the total to one row or one column. That is conditional probability.

## The key idea

$$
P(A) = \frac{\text{outcomes in } A}{\text{all outcomes}} \qquad P(A \mid B) = \frac{\text{outcomes in both } A \text{ and } B}{\text{outcomes in } B}
$$

Read \(P(A \mid B)\) as "the probability of A, given B". You only look at the group B, so B's count goes on the bottom.

**Survey of 200 students: do you play a musical instrument?**

|  | Plays | Does not play | Total |
| --- | --- | --- | --- |
| Grade 10 | 36 | 64 | 100 |
| Grade 11 | 54 | 46 | 100 |
| Total | 90 | 110 | 200 |

## Worked examples

**Example 1: a simple probability**

Problem: Using the survey table, one student is chosen at random. What is the probability the student plays an instrument?

1. The group is all 200 students. 90 of them play.

   $$
   \frac{90}{200} = \frac{9}{20}
   $$

Answer: \(\frac{9}{20}\), or 0.45

**Example 2: given a row**

Problem: A grade 11 student is chosen at random. What is the probability the student plays an instrument?

1. "A grade 11 student" means the total is the grade 11 row, 100 students. 54 of them play.

   $$
   \frac{54}{100} = \frac{27}{50}
   $$

Answer: \(\frac{27}{50}\), or 0.54

**Example 3: the same numbers, the other condition**

Problem: A student who plays an instrument is chosen at random. What is the probability the student is in grade 11?

1. Now the group is the "Plays" column, 90 students. 54 of them are in grade 11.

   $$
   \frac{54}{90} = \frac{3}{5}
   $$
2. Compare with Example 2. The top number is the same, 54, but the total changed. That is the whole trick of conditional probability.

Answer: \(\frac{3}{5}\), or 0.6

**Example 4 (SAT-hard): two draws without replacement**

Problem: A bag has 4 red marbles and 6 blue marbles. Two marbles are drawn at random, one after the other, without putting the first back. What is the probability both are red?

1. First draw: 4 red out of 10 marbles.

   $$
   \frac{4}{10}
   $$
2. Second draw: one red marble is gone, so 3 red out of 9.

   $$
   \frac{3}{9}
   $$
3. Both must happen, so multiply.

   $$
   \frac{4}{10} \cdot \frac{3}{9} = \frac{12}{90} = \frac{2}{15}
   $$

Answer: \(\frac{2}{15}\)

## Common mistakes

- **Using the grand total for a conditional question.** "Of the grade 11 students" means divide by 100, not 200. Fix: underline the group the question chooses from.
- **Swapping the condition.** \(P(\text{plays} \mid \text{grade 11})\) is \(\frac{54}{100}\), but \(P(\text{grade 11} \mid \text{plays})\) is \(\frac{54}{90}\). Fix: the condition after "given" is the bottom number.
- **Forgetting the first draw changes the bag.** Without replacement, the second draw has one fewer marble. Fix: update both the top and bottom after each draw.
- **Adding when you should multiply.** "Both" or "and" for separate events means multiply. Fix: add only when you combine groups that do not overlap, like red or blue.
- **Giving a probability above 1.** If you get 1.8, a total is wrong. Fix: the top is always part of the bottom.

## Quick method

> **Tip: Circle the row or column**
>
> Before computing, circle the row or column the question limits you to. Its total is your denominator. This one habit prevents most two-way table mistakes.

## Practice

Questions 1 to 3 use this table.

**400 store orders by payment and pickup**

|  | In store | Online pickup | Total |
| --- | --- | --- | --- |
| Card | 150 | 90 | 240 |
| Cash | 130 | 30 | 160 |
| Total | 280 | 120 | 400 |

**5 SAT-style questions**

1. One order is chosen at random. What is the probability it was paid with cash?
   A. \(\frac{13}{40}\)
   B. \(\frac{1}{4}\)
   C. \(\frac{2}{5}\)
   D. \(\frac{3}{5}\)

   Answer: \(\frac{2}{5}\). 160 of the 400 orders were cash: \(\frac{160}{400} = \frac{2}{5}\). \(\frac{13}{40}\) counts only cash orders in store, and \(\frac{3}{5}\) is card.

2. An order paid with a card is chosen at random. What is the probability it was an online pickup?
   A. \(\frac{9}{40}\)
   B. \(\frac{3}{8}\)
   C. \(\frac{3}{5}\)
   D. \(\frac{3}{4}\)

   Answer: \(\frac{3}{8}\). The group is card orders, 240. 90 were pickups: \(\frac{90}{240} = \frac{3}{8}\). \(\frac{9}{40}\) uses all 400 orders, and \(\frac{3}{4}\) answers the reverse question.

3. An online pickup order is chosen at random. What is the probability it was paid with a card?
   A. \(\frac{1}{4}\)
   B. \(\frac{9}{40}\)
   C. \(\frac{3}{8}\)
   D. \(\frac{3}{4}\)

   Answer: \(\frac{3}{4}\). The group is pickups, 120. 90 used a card: \(\frac{90}{120} = \frac{3}{4}\). \(\frac{1}{4}\) is the cash share of pickups.

4. Two fair six-sided dice are rolled. What is the probability that the sum is 8?
   A. \(\frac{1}{9}\)
   B. \(\frac{5}{36}\)
   C. \(\frac{1}{6}\)
   D. \(\frac{2}{9}\)

   Answer: \(\frac{5}{36}\). There are 36 equally likely rolls. A sum of 8 comes from (2, 6), (3, 5), (4, 4), (5, 3) and (6, 2): 5 rolls. \(\frac{1}{6}\) is the chance of a sum of 7.

5. Student-produced response: a spinner has 5 equal sections numbered 1 to 5. It is spun twice. What is the probability that both spins land on an even number?

   Answer: 4/25. Two of the 5 numbers are even, so each spin is \(\frac{2}{5}\). The spins do not affect each other, so multiply: \(\frac{2}{5} \cdot \frac{2}{5} = \frac{4}{25}\). You could also enter .16.

## Frequently asked questions

### What does "given that" mean in probability?

It means you only look at one group. "The probability a student plays an instrument, given that the student is in grade 11" uses only grade 11 students as the total. That is conditional probability, written \(P(A \mid B)\).

### How do I read a two-way table quickly?

Find the group the question chooses from; its total is the bottom of your fraction. Then find the cell where that group meets the trait you want; that is the top. Use the grand total only when the question picks from everyone.

### Can I enter a probability as a fraction on the SAT?

Yes. College Board's directions allow fractions like 4/25 or decimals like .16 on student-produced responses. If a fraction is too long for the 5-character box, enter the decimal instead.

## Sources

- [College Board: SAT Math, Problem-Solving and Data Analysis skills (probability and conditional probability)](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/problem-solving), accessed 2026-10-01
- [College Board: SAT Bluebook test directions (student-produced responses)](https://satsuite.collegeboard.org/media/pdf/english-sat-test-directions-bb.pdf), accessed 2026-10-01

## Related

- [Mean, median and spread on the SAT](https://duckyhelper.com/learn/sat-math/statistics/)
- [Percentages on the SAT](https://duckyhelper.com/learn/sat-math/percentages/)
- [Scatterplots, margin of error and study design on the SAT](https://duckyhelper.com/learn/sat-math/scatterplots-and-inference/)
- [SAT Math study guides](https://duckyhelper.com/learn/sat-math/)

## Try asking Ducky

- "Which number goes on the bottom for this one?"
- "Why is it 54 over 90 and not 54 over 100?"
- "Make me a two-way table and ask me three questions about it."

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