Probability and Two-Way Tables

Calculate probabilities from two-way tables and counting principles.

Written and reviewed by the Study Mondo Education TeamLast updated August 17, 2026

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Probability and Two-Way Tables on the SAT

Basic Probability

$\text{Probability of event} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

Probability is always between 0 and 1 (or 0% and 100%)
$P = 0$ : impossible
$P = 1$ : certain

Two-Way Tables (Contingency Tables)

A two-way table organizes data by two categorical variables.

Example:

	Likes Pizza	Doesn't Like Pizza	Total
Students	45	15	60
Teachers	20	10	30
Total	65	25	90

Reading the Table

Row totals are on the right
Column totals are on the bottom
Grand total is bottom-right

Types of Probability from Two-Way Tables

Joint Probability

The probability of two specific categories together.

$\text{Probability of Student AND Likes Pizza} = \frac{45}{90} = \frac{1}{2}$

The denominator is the grand total.

Marginal Probability

The probability of just one category.

$\text{Probability of Student} = \frac{60}{90} = \frac{2}{3}$

$\text{Probability of Likes Pizza} = \frac{65}{90} = \frac{13}{18}$

Conditional Probability

The probability of one event GIVEN another has occurred.

Selected from the students: probability of liking pizza $= \frac{45}{60} = \frac{3}{4}$

Key: The denominator is the subtotal of the given condition, not the grand total!

Selected from those who like pizza: probability of being a student $= \frac{45}{65} = \frac{9}{13}$

Conditional Probability Formula

The "from" rule: the group named after "from" (or "given") becomes your denominator — its row or column total, not the grand total. The SAT always words it this way; you will not see formal notation on the test.

The SAT usually tests this with two-way tables rather than the formula directly.

Complement Rule

$P(\text{NOT } A) = 1 - P(A)$

If the probability of rain is 0.3, the probability of no rain is $1 - 0.3 = 0.7$ .

SAT Question Types

Type 1: "What is the probability that a randomly selected person...?"

Identify the numerator (favorable outcomes) and denominator (total)
Watch for whether it's conditional ("...given that they are a student")

Type 2: "What fraction of [group] are [category]?"

This is conditional probability. The denominator is the SIZE of the given group.

Type 3: "Which group has a higher proportion of...?"

Compare conditional probabilities between groups.

Type 4: Complete a Two-Way Table

Fill in missing values using row/column totals. Every row and column must add up.

Common SAT Mistakes

Using the wrong denominator — the #1 mistake! For conditional probability, use the row/column total, NOT the grand total
Confusing "and" with "given/from" — "junior AND walks" uses the grand total; "selected from the juniors" uses the junior total
Misreading which row/column represents which category
Not simplifying fractions when answer choices are simplified
Forgetting the complement — sometimes it's easier to calculate $1 - P(\text{not happening})$

📚 Practice Problems

1Problem 1easy

❓ Question:

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly selecting a blue marble?

💡 Show Solution

Step 1: Count total marbles: $5 + 3 + 2 = 10$

Step 2: Apply the probability formula: $\text{Probability of blue} = \frac{\text{blue marbles}}{\text{total marbles}} = \frac{3}{10}$$

Answer: $\frac{3}{10}$ or $0.3$ or $30\%$

2Problem 2easy

❓ Question:

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly selecting a blue marble?

💡 Show Solution

Step 1: Count total marbles: $5 + 3 + 2 = 10$

Step 2: Apply the probability formula: $\text{Probability of blue} = \frac{\text{blue marbles}}{\text{total marbles}} = \frac{3}{10}$$

Answer: $\frac{3}{10}$ or $0.3$ or $30\%$

3Problem 3easy

❓ Question:

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly selecting a blue marble?

💡 Show Solution

Step 1: Count total marbles: $5 + 3 + 2 = 10$

Step 2: Apply the probability formula: $\text{Probability of blue} = \frac{\text{blue marbles}}{\text{total marbles}} = \frac{3}{10}$$

Answer: $\frac{3}{10}$ or $0.3$ or $30\%$

4Problem 4medium

❓ Question:

Use the table below:

	Passed	Failed	Total
Studied	42	8	50
Did Not Study	18	32	50
Total	60	40	100

What is the probability that a student passed, given that they studied?

💡 Show Solution

Key: This is a conditional probability question because of the phrase "given that they studied."

Step 1: Identify the condition: "given that they studied" means we only look at the "Studied" row.

Step 2: The denominator is the total who studied: 50 The numerator is those who studied AND passed: 42

$\text{From those who studied: } \frac{42}{50} = \frac{21}{25} = 0.84$

Answer: $\frac{21}{25}$ or $84\%$

Common mistake: Using 100 as the denominator (that would give the joint probability, not the conditional probability).

5Problem 5medium

❓ Question:

Use the table below:

	Passed	Failed	Total
Studied	42	8	50
Did Not Study	18	32	50
Total	60	40	100

What is the probability that a student passed, given that they studied?

💡 Show Solution

Key: This is a conditional probability question because of the phrase "given that they studied."

Step 1: Identify the condition: "given that they studied" means we only look at the "Studied" row.

Step 2: The denominator is the total who studied: 50 The numerator is those who studied AND passed: 42

$\text{From those who studied: } \frac{42}{50} = \frac{21}{25} = 0.84$

Answer: $\frac{21}{25}$ or $84\%$

Common mistake: Using 100 as the denominator (that would give the joint probability, not the conditional probability).

6Problem 6medium

❓ Question:

Use the table below:

	Passed	Failed	Total
Studied	42	8	50
Did Not Study	18	32	50
Total	60	40	100

What is the probability that a student passed, given that they studied?

💡 Show Solution

Key: This is a conditional probability question because of the phrase "given that they studied."

Step 1: Identify the condition: "given that they studied" means we only look at the "Studied" row.

Step 2: The denominator is the total who studied: 50 The numerator is those who studied AND passed: 42

$\text{From those who studied: } \frac{42}{50} = \frac{21}{25} = 0.84$

Answer: $\frac{21}{25}$ or $84\%$

Common mistake: Using 100 as the denominator (that would give the joint probability, not the conditional probability).

7Problem 7medium

❓ Question:

Using the same table above, what fraction of students who passed had studied?

💡 Show Solution

Key: This question asks "of those who passed" — so the condition is passing.

Step 1: The denominator is the total who passed: 60 The numerator is those who passed AND studied: 42

$\text{From those who passed: } \frac{42}{60} = \frac{7}{10}$

Answer: $\frac{7}{10}$

Important: Notice this is DIFFERENT from the previous question! $P(\text{Passed} | \text{Studied}) = \frac{42}{50}$ but $P(\text{Studied} | \text{Passed}) = \frac{42}{60}$ . The order matters in conditional probability!

8Problem 8medium

❓ Question:

Using the same table above, what fraction of students who passed had studied?

💡 Show Solution

Key: This question asks "of those who passed" — so the condition is passing.

Step 1: The denominator is the total who passed: 60 The numerator is those who passed AND studied: 42

$\text{From those who passed: } \frac{42}{60} = \frac{7}{10}$

Answer: $\frac{7}{10}$

9Problem 9medium

❓ Question:

Using the same table above, what fraction of students who passed had studied?

💡 Show Solution

Key: This question asks "of those who passed" — so the condition is passing.

Step 1: The denominator is the total who passed: 60 The numerator is those who passed AND studied: 42

$\text{From those who passed: } \frac{42}{60} = \frac{7}{10}$

Answer: $\frac{7}{10}$

10Problem 10hard

❓ Question:

A survey asked 200 people about their exercise habits and diet:

	Exercises Regularly	Does Not Exercise	Total
Healthy Diet	65	?	100
Unhealthy Diet	?	60	?
Total	?	?	200

Complete the table and find the probability that a randomly selected person exercises regularly OR has a healthy diet.

💡 Show Solution

Step 1: Fill in the table.

Healthy Diet row: Does Not Exercise = $100 - 65 = 35$ Unhealthy Diet total = $200 - 100 = 100$ Unhealthy Diet, Exercises = $100 - 60 = 40$ Exercises total = $65 + 40 = 105$ Does Not Exercise total = $35 + 60 = 95$

Completed table:

	Exercises	Doesn't	Total
Healthy	65	35	100
Unhealthy	40	60	100
Total	105	95	200

Step 2: Find $P(\text{Exercises OR Healthy Diet})$

Use the inclusion-exclusion principle: Counting 'A or B': add the two groups, then subtract the overlap once (it was counted twice). $= \frac{105}{200} + \frac{100}{200} - \frac{65}{200} = \frac{140}{200} = \frac{7}{10}$

Answer: $\frac{7}{10}$ or $70\%$

11Problem 11hard

❓ Question:

A survey asked 200 people about their exercise habits and diet:

	Exercises Regularly	Does Not Exercise	Total
Healthy Diet	65	?	100
Unhealthy Diet	?	60	?
Total	?	?	200

Complete the table and find the probability that a randomly selected person exercises regularly OR has a healthy diet.

💡 Show Solution

Step 1: Fill in the table.

Completed table:

	Exercises	Doesn't	Total
Healthy	65	35	100
Unhealthy	40	60	100
Total	105	95	200

Step 2: Find $P(\text{Exercises OR Healthy Diet})$

Answer: $\frac{7}{10}$ or $70\%$

12Problem 12hard

❓ Question:

A survey asked 200 people about their exercise habits and diet:

	Exercises Regularly	Does Not Exercise	Total
Healthy Diet	65	?	100
Unhealthy Diet	?	60	?
Total	?	?	200

Complete the table and find the probability that a randomly selected person exercises regularly OR has a healthy diet.

💡 Show Solution

Step 1: Fill in the table.

Completed table:

	Exercises	Doesn't	Total
Healthy	65	35	100
Unhealthy	40	60	100
Total	105	95	200

Step 2: Find $P(\text{Exercises OR Healthy Diet})$

Answer: $\frac{7}{10}$ or $70\%$

13Problem 13expert

❓ Question:

In a class, the probability of a student playing basketball is 0.4, the probability of playing soccer is 0.3, and the probability of playing both is 0.1. What is the probability that a randomly chosen student plays basketball but NOT soccer?

💡 Show Solution

Step 1: Use the relationship: $\text{Probability of Basketball only} = P(\text{Basketball}) - P(\text{Basketball AND Soccer})= 0.4 - 0.1 = 0.3$$

Step 2: Verify with a Venn diagram mental model:

Basketball only: 0.3
Soccer only: $0.3 - 0.1 = 0.2$
Both: 0.1
Neither: $1 - (0.3 + 0.2 + 0.1) = 0.4$

All probabilities sum to 1: $0.3 + 0.2 + 0.1 + 0.4 = 1$ ✓

Answer: $0.3$ or $30\%$

SAT Tip: "A but NOT B" means subtract the overlap from A's probability.

14Problem 14expert

❓ Question:

💡 Show Solution

Step 1: Use the relationship: $\text{Probability of Basketball only} = P(\text{Basketball}) - P(\text{Basketball AND Soccer})= 0.4 - 0.1 = 0.3$$

Step 2: Verify with a Venn diagram mental model:

Basketball only: 0.3
Soccer only: $0.3 - 0.1 = 0.2$
Both: 0.1
Neither: $1 - (0.3 + 0.2 + 0.1) = 0.4$

All probabilities sum to 1: $0.3 + 0.2 + 0.1 + 0.4 = 1$ ✓

Answer: $0.3$ or $30\%$

SAT Tip: "A but NOT B" means subtract the overlap from A's probability.

15Problem 15expert

❓ Question:

💡 Show Solution

Step 1: Use the relationship: $\text{Probability of Basketball only} = P(\text{Basketball}) - P(\text{Basketball AND Soccer})= 0.4 - 0.1 = 0.3$$

Step 2: Verify with a Venn diagram mental model:

Basketball only: 0.3
Soccer only: $0.3 - 0.1 = 0.2$
Both: 0.1
Neither: $1 - (0.3 + 0.2 + 0.1) = 0.4$

All probabilities sum to 1: $0.3 + 0.2 + 0.1 + 0.4 = 1$ ✓

Answer: $0.3$ or $30\%$

SAT Tip: "A but NOT B" means subtract the overlap from A's probability.

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What is Probability and Two-Way Tables?▾

Calculate probabilities from two-way tables and counting principles.

How can I study Probability and Two-Way Tables effectively?▾

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Yes, this page includes 15 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Probability and Two-Way Tables

Try the Interactive Version!

Probability and Two-Way Tables on the SAT

Basic Probability

Two-Way Tables (Contingency Tables)

Reading the Table

Types of Probability from Two-Way Tables

Joint Probability

Marginal Probability

Conditional Probability

Conditional Probability Formula

Complement Rule

SAT Question Types

Type 1: "What is the probability that a randomly selected person...?"

Type 2: "What fraction of [group] are [category]?"

Type 3: "Which group has a higher proportion of...?"

Type 4: Complete a Two-Way Table

Common SAT Mistakes

📚 Practice Problems

1Problem 1easy

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2Problem 2easy

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3Problem 3easy

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4Problem 4medium

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5Problem 5medium

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6Problem 6medium

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7Problem 7medium

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9Problem 9medium

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10Problem 10hard

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11Problem 11hard

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12Problem 12hard

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13Problem 13expert

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14Problem 14expert

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15Problem 15expert

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❓ Frequently Asked Questions