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Calculate probabilities from two-way tables and counting principles.
Learn step-by-step with practice exercises built right in.
$\text{Probability of event} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$
A two-way table organizes data by two categorical variables.
Example:
| Likes Pizza | Doesn't Like Pizza | Total | |
|---|---|---|---|
| Students | 45 | 15 | 60 |
| Teachers | 20 | 10 | 30 |
| Total | 65 | 25 | 90 |
The probability of two specific categories together.
The denominator is the grand total.
The probability of just one category.
The probability of one event GIVEN another has occurred.
Selected from the students: probability of liking pizza
Key: The denominator is the subtotal of the given condition, not the grand total!
Selected from those who like pizza: probability of being a student
The "from" rule: the group named after "from" (or "given") becomes your denominator — its row or column total, not the grand total. The SAT always words it this way; you will not see formal notation on the test.
The SAT usually tests this with two-way tables rather than the formula directly.
If the probability of rain is 0.3, the probability of no rain is .
This is conditional probability. The denominator is the SIZE of the given group.
Compare conditional probabilities between groups.
Fill in missing values using row/column totals. Every row and column must add up.
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly selecting a blue marble?
Step 1: Count total marbles:
Step 2: Apply the probability formula: $\text{Probability of blue} = \frac{\text{blue marbles}}{\text{total marbles}} = \frac{3}{10}$$
Answer: or or
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly selecting a blue marble?
Step 1: Count total marbles:
Step 2: Apply the probability formula: $\text{Probability of blue} = \frac{\text{blue marbles}}{\text{total marbles}} = \frac{3}{10}$$
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly selecting a blue marble?
Step 1: Count total marbles:
Step 2: Apply the probability formula: $\text{Probability of blue} = \frac{\text{blue marbles}}{\text{total marbles}} = \frac{3}{10}$$
Use the table below:
| Passed | Failed | Total | |
|---|---|---|---|
| Studied | 42 | 8 | 50 |
| Did Not Study | 18 | 32 | 50 |
| Total | 60 |
Use the table below:
| Passed | Failed | Total | |
|---|---|---|---|
| Studied | 42 | 8 | 50 |
| Did Not Study | 18 | 32 | 50 |
| Total | 60 |
Use the table below:
| Passed | Failed | Total | |
|---|---|---|---|
| Studied | 42 | 8 | 50 |
| Did Not Study | 18 | 32 | 50 |
| Total | 60 |
Using the same table above, what fraction of students who passed had studied?
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
Using the same table above, what fraction of students who passed had studied?
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
Using the same table above, what fraction of students who passed had studied?
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
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 | ? |
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 | ? |
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 | ? |
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?
Step 1: Use the relationship: $\text{Probability of Basketball only} = P(\text{Basketball}) - P(\text{Basketball AND Soccer})= 0.4 - 0.1 = 0.3$$
Step 2: Verify with a Venn diagram mental model:
In a class, the probability of a student playing basketball is 0.4, the probability of playing soccer is 0.3, and the probability of playing both is 0.1. What is the probability that a randomly chosen student plays basketball but NOT soccer?
Step 1: Use the relationship: $\text{Probability of Basketball only} = P(\text{Basketball}) - P(\text{Basketball AND Soccer})= 0.4 - 0.1 = 0.3$$
Step 2: Verify with a Venn diagram mental model:
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?
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:
Answer: or or
Answer: or or
| 40 |
| 100 |
What is the probability that a student passed, given that they studied?
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
Answer: or
Common mistake: Using 100 as the denominator (that would give the joint probability, not the conditional probability).
| 40 |
| 100 |
What is the probability that a student passed, given that they studied?
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
Answer: or
Common mistake: Using 100 as the denominator (that would give the joint probability, not the conditional probability).
| 40 |
| 100 |
What is the probability that a student passed, given that they studied?
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
Answer: or
Common mistake: Using 100 as the denominator (that would give the joint probability, not the conditional probability).
Answer:
Important: Notice this is DIFFERENT from the previous question! but . The order matters in conditional probability!
Answer:
Important: Notice this is DIFFERENT from the previous question! but . The order matters in conditional probability!
Answer:
Important: Notice this is DIFFERENT from the previous question! but . The order matters in conditional probability!
| ? |
| 200 |
Complete the table and find the probability that a randomly selected person exercises regularly OR has a healthy diet.
Step 1: Fill in the table.
Healthy Diet row: Does Not Exercise = Unhealthy Diet total = Unhealthy Diet, Exercises = Exercises total = Does Not Exercise total =
Completed table:
| Exercises | Doesn't | Total | |
|---|---|---|---|
| Healthy | 65 | 35 | 100 |
| Unhealthy | 40 | 60 | 100 |
| Total | 105 | 95 | 200 |
Step 2: Find
Use the inclusion-exclusion principle: Counting 'A or B': add the two groups, then subtract the overlap once (it was counted twice).
Answer: or
| ? |
| 200 |
Complete the table and find the probability that a randomly selected person exercises regularly OR has a healthy diet.
Step 1: Fill in the table.
Healthy Diet row: Does Not Exercise = Unhealthy Diet total = Unhealthy Diet, Exercises = Exercises total = Does Not Exercise total =
Completed table:
| Exercises | Doesn't | Total | |
|---|---|---|---|
| Healthy | 65 | 35 | 100 |
| Unhealthy | 40 | 60 | 100 |
| Total | 105 | 95 | 200 |
Step 2: Find
Use the inclusion-exclusion principle: Counting 'A or B': add the two groups, then subtract the overlap once (it was counted twice).
Answer: or
| ? |
| 200 |
Complete the table and find the probability that a randomly selected person exercises regularly OR has a healthy diet.
Step 1: Fill in the table.
Healthy Diet row: Does Not Exercise = Unhealthy Diet total = Unhealthy Diet, Exercises = Exercises total = Does Not Exercise total =
Completed table:
| Exercises | Doesn't | Total | |
|---|---|---|---|
| Healthy | 65 | 35 | 100 |
| Unhealthy | 40 | 60 | 100 |
| Total | 105 | 95 | 200 |
Step 2: Find
Use the inclusion-exclusion principle: Counting 'A or B': add the two groups, then subtract the overlap once (it was counted twice).
Answer: or
All probabilities sum to 1: ✓
Answer: or
SAT Tip: "A but NOT B" means subtract the overlap from A's probability.
All probabilities sum to 1: ✓
Answer: or
SAT Tip: "A but NOT B" means subtract the overlap from A's probability.
All probabilities sum to 1: ✓
Answer: or
SAT Tip: "A but NOT B" means subtract the overlap from A's probability.