Two-Way Tables and Conditional Probability

What is a Two-Way Table?

A two-way table (also called a contingency table) shows the relationship between two categorical variables.

Example: Survey of 100 students about pets:

| | Dog | Cat | Total | |-----------|-----|-----|-------| | Male | 24 | 16 | 40 | | Female | 36 | 24 | 60 | | Total | 60 | 40 | 100 |

Reading Two-Way Tables

Key vocabulary:

• Joint frequency: Cell value (e.g., 24 males have dogs)
• Marginal frequency: Row/column total (e.g., 40 males total)
• Total: Overall sum (100 students)

Conditional Probability

Formula: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$

Read as: "Probability of A given B"

Example Questions

Q1: What's the probability a randomly selected student is female? $P(\text{Female}) = \frac{60}{100} = 0.6$

Q2: What's the probability a student has a dog, given they're male? $P(\text{Dog}|\text{Male}) = \frac{24}{40} = 0.6$

Q3: What's the probability a student is male, given they have a cat? $P(\text{Male}|\text{Cat}) = \frac{16}{40} = 0.4$

SAT Strategy

Step 1: Identify What You're Finding

Look for "given" or "if" → use conditional probability

"Given that a student is female, what's the probability they have a cat?"

• Condition: Female (use Female row only)
• Want: Cat
• Answer: $\frac{24}{60} = 0.4$

Step 2: Find the Right Row or Column

The condition determines your denominator:

• "Given male" → Use male row (denominator = 40)
• "Given has dog" → Use dog column (denominator = 60)

Step 3: Calculate

$\text{Conditional Probability} = \frac{\text{Intersection}}{\text{Condition Total}}$

Common SAT Mistakes

❌ Using the wrong denominator (using 100 instead of the condition total)
❌ Confusing $P(A|B)$ with $P(B|A)$ — order matters!
❌ Not identifying the condition properly
❌ Adding when you should divide

Practice Tip

Always ask: "Out of WHAT?" This tells you the denominator.

• "Out of all males" → denominator is 40
• "Out of all students" → denominator is 100
• "Out of all dog owners" → denominator is 60