Probability and Two-Way Tables

Problem Solving: Ratios, Rates & Proportions

Part 1 of 7 — Setting Up and Solving Proportions

This is one of the most heavily tested topics on the SAT Math section. About 25-30% of Math questions fall under Problem Solving & Data Analysis.

Ratios

A ratio compares two quantities: If a recipe uses 3 cups flour to 2 cups sugar, the ratio is 3:2 or 3/2.

Setting Up Proportions

Cross-multiply to solve:

$\frac{3}{5} = \frac{x}{20} \implies 3 \times 20 = 5x \implies x = 12$

Unit Rates

A unit rate has a denominator of 1:

• 240 miles in 4 hours → 60 mph
• $45 for 3 shirts → $15 per shirt

SAT Trap: Mixing Up Parts and Wholes

If the ratio of boys to girls is 3:5, there are 8 total parts (not 5).

• Boys = 3/8 of total
• Girls = 5/8 of total

Dimensional Analysis

Convert units by multiplying fractions: $60 \frac{\text{miles}}{\text{hour}} \times \frac{1 \text{ hour}}{60 \text{ min}} = 1 \frac{\text{mile}}{\text{min}}$

Deep Dive: Complex Ratio & Proportion Problems

Worked Example 1: Multi-Step Ratio

Worked Example 2: Unit Conversion Chain

Ratio vs. Fraction — Key Difference

Dimensional Analysis — Multi-Step

Convert 45 mph to feet per second:

$45 \frac{\text{mi}}{\text{hr}} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ sec}} = 66 \frac{\text{ft}}{\text{sec}}$

Advanced Ratio & Proportion Problems 🎯

Ratio & Proportion Setup — Select the correct approach.

Part 1 Summary: Ratios, Rates & Proportions

Next: Percentages — increase, decrease, and successive changes →

Percentages: Increase, Decrease & Applications

Part 2 of 7 — Mastering Percent Problems

Percent Formula

$\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100$

Percent Increase/Decrease

$\text{\% Change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100$

Shortcut multipliers:

• 20% increase → multiply by 1.20
• 15% decrease → multiply by 0.85
• 8% tax → multiply by 1.08

Successive Percent Changes

A 10% increase followed by a 10% decrease is NOT back to the original: $100 \xrightarrow{+10\%} 110 \xrightarrow{-10\%} 99$

SAT Classic: "What percent of X is Y?"

Translate directly: "What percent of 80 is 24?" $\frac{24}{80} \times 100 = 30\%$

Percent vs. Percentage Points

"Increased from 40% to 52%" = increase of 12 percentage points but a 30% increase (12/40 \times 100).

Deep Dive: Percent Problem Strategies

Worked Example 1: Finding Original Price

Worked Example 2: Successive Changes

Percent Multiplier Quick Reference

"Percent OF" vs. "Percent MORE THAN"

• "A is 25% of B" → $A = 0.25B$
• "A is 25% more than B" → $A = 1.25B$
• "A is 25% less than B" → $A = 0.75B$

Advanced Percent Problems 🎯

Percent Multiplier Check — Select the correct multiplier.

Part 2 Summary: Percentages

SAT Traps

• Successive equal percent changes DON'T cancel out
• "A is 60% more than B" ≠ "B is 60% less than A"
• Always divide by the original for percent change

Next: Two-way tables and data interpretation →

Two-Way Tables & Data Interpretation

Part 3 of 7 — Reading Tables and Finding Probabilities

Two-Way Tables

These organize data by two categories. Example:

Fractions of One Group

"What fraction of sophomores are female?"

• Look at the Sophomore column: 150 female out of 250 total = 150/250 = 3/5

"Selected From" = Restrict to a Subgroup

"If a student is selected at random from the males, what is the probability the student is a freshman?"

• Restrict to the Male row: 120 freshmen out of 220 males = 120/220 = 6/11

Everyone vs. One Group

• Selected from everyone: the probability of a female is 280/500 — uses the grand total
• Selected from the sophomores: the probability of a female is 150/250 — the named group's total is the denominator

Is There an Association?

Compare each group's rate, in words.

• If freshmen and sophomores are female at the same rate, the data show no association between class and gender
• If the rates differ, the data suggest an association
• Here: freshmen are $130/250 = 52\%$ female and sophomores are $150/250 = 60\%$ female, so the data suggest an association

Deep Dive: Navigating Two-Way Tables

Worked Example 1: Filling In a Table

Worked Example 2: Checking for an Association

Denominator Guide

SAT Trap: Joint vs. Conditional

• Selected from everyone: male AND freshman $= 120/500$ (out of everyone)
• Selected from the males: freshman $= 120/220$ (males only)

Advanced Two-Way Table Problems 🎯

Pick the Right Denominator — What goes in the denominator for each question?

Part 3 Summary: Two-Way Tables

SAT Strategy

• Read the question word-for-word to find the correct denominator.
• "Selected from" or "among" a group → use that group's subtotal.
• "Of all" = marginal → use the grand total.

Next: Statistics — mean, median, and standard deviation →

Statistics: Center, Spread & Shape

Part 4 of 7 — Mean, Median, Standard Deviation

Measures of Center

• Mean = sum of all values / count. Sensitive to outliers.
• Median = middle value when sorted. Resistant to outliers.

When to Use Mean vs. Median

• Symmetric data → mean ≈ median, use either
• Skewed data or outliers → median is more representative

Standard Deviation

Measures how spread out data is from the mean.

• Low SD → data points close to mean (consistent)
• High SD → data points far from mean (variable)

You won't calculate SD on the SAT, but you must compare SDs:

• {10, 10, 10, 10, 10} → SD = 0 (no spread)
• {8, 9, 10, 11, 12} → small SD
• {1, 3, 10, 17, 19} → large SD

Effect of Adding/Removing Values

• Adding a value equal to the mean → mean unchanged, SD decreases
• Adding an outlier → mean shifts toward outlier, SD increases
• Removing an outlier → mean moves away from outlier, SD decreases

Shape of Distributions

• Right-skewed (tail to right): mean > median
• Left-skewed (tail to left): mean < median
• Symmetric: mean ≈ median

Deep Dive: Statistics in Action

Worked Example 1: Finding a Missing Value

Worked Example 2: Effect of Removing a Value

Adding a Constant vs. Multiplying

SAT favorite: "If every student's score increases by 5 points, what happens to the standard deviation?" → Nothing — adding a constant shifts all values equally.

Skewness Quick Reference

Advanced Statistics Problems 🎯

Statistics Quick Check — Select the correct answer.

Part 4 Summary: Statistics

Key Rules

• Add constant $k$: mean & median shift by $k$, SD unchanged
• Multiply by $k$: mean, median, & SD all multiply by $|k|$
• Right-skewed → mean $>$ median
• Outlier → use median as the better center

Next: Scatterplots and line of best fit →

Scatterplots & Line of Best Fit

Part 5 of 7 — Interpreting Trends and Making Predictions

Reading Scatterplots

• Positive association: as x increases, y increases (upward trend)
• Negative association: as x increases, y decreases (downward trend)
• No association: no visible pattern

Line/Curve of Best Fit

The line that follows the overall trend of the points as closely as possible. Key interpretations:

• Slope = rate of change (For each 1-unit increase in x, y changes by [slope])
• y-intercept = predicted y-value when x = 0

Making Predictions

Use the equation to predict values:

• If y = 2.3x + 15 models study hours vs. test score:
• 10 hours → predicted score: 2.3(10) + 15 = 38

Interpolation vs. Extrapolation

• Interpolation (within data range): reliable predictions
• Extrapolation (beyond data range): unreliable — the trend may not continue

Residuals

Residual = actual – predicted

• Positive residual: actual is above the line
• Negative residual: actual is below the line
• Random residuals → good model
• Patterned residuals (curved) → wrong model type

Deep Dive: Scatterplot Analysis

Worked Example 1: Interpreting Slope in Context

Worked Example 2: Choosing the Best Model

Worked Example 3: Describing an Association in Words

The SAT describes an association in words: direction, form, and strength.

SAT key fact: strength is how closely the points follow the trend, not how steep the trend is. A steep line with widely scattered points is a weak association; a gentle line that the points hug is a strong one.

Advanced Scatterplot Problems 🎯

Scatterplot Interpretation — Select the correct answer.

Part 5 Summary: Scatterplots & Best Fit

Next: Probability →

Probability & Predictions

Part 6 of 7 — SAT Probability Essentials

The digital SAT keeps probability concrete: counts, tables, and proportions, always in plain words. You will never see formal notation like P(A|B) or union/intersection symbols on the test.

Basic Probability

$\text{Probability} = \frac{\text{favorable outcomes}}{\text{total outcomes}}$

"NOT" Questions

The probability something does NOT happen is 1 minus the probability it does — or just count the non-favorable outcomes directly.

Example: 4 red, 6 blue, 5 green marbles. The probability of NOT drawing red — count the 11 non-red marbles: $\frac{11}{15}$.

The Three SAT Probability Setups

1. From a table: "If a student is selected at random from those who…"
2. From counts: "A bag holds 3 red and 5 blue marbles…"
3. From a survey: "Based on the results, what proportion…"

Predicting a Count ("how many would you expect")

$\text{Expected count} = \text{Total} \times \text{proportion}$

Example: 200 people surveyed, 35% prefer A → expect $200 \times 0.35 = 70$ of the next 200 to prefer A.

Relative Frequency

Just another word for proportion: $\text{Relative frequency of A} = \frac{\text{count of A}}{\text{total count}}$

Deep Dive: Table Probability — the SAT's Favorite Question

Worked Example 1: One Cell Over the Grand Total

"If a student is selected at random from all 80, what is the probability the student attended the review AND passed?"

One cell over the grand total: $\frac{32}{80} = \frac{2}{5}$.

Worked Example 2: The "From" Rule (conditional, in words)

"If a student is selected at random from those who SKIPPED the review, what is the probability the student passed?"

"From those who skipped" restricts you to that row: $\frac{18}{40} = \frac{9}{20}$.

The word after "from" names your denominator. This is exactly how the SAT asks conditional probability — no notation, just words.

Worked Example 3: Predicting a Count

"In a random sample, 42% prefer Brand A. If 500 people are surveyed from the same population, how many would you expect to prefer Brand A?"

$500 \times 0.42 = 210$ people.

SAT Probability from Tables — the Full Playbook

1. No restriction ("from all participants"): cell \div grand total
2. Restricted group ("from the seniors" / "from those who said yes"): cell \div that row or column total
3. "NOT": count the other cells, or use 1 minus

Table Probability Problems 🎯

Pick the Right Denominator — Match each question to the correct setup.

Part 6 Summary: Probability

• The word "from" names your denominator — that's the whole skill
• The SAT never uses P(A|B) or ∪/∩ symbols — everything is words and tables

Next: Comprehensive review and mixed practice →

Problem Solving & Data Review

Part 7 of 7 — Mixed Practice & Strategy

Topic Checklist

✓ Ratios, rates, proportions, and unit conversion ✓ Percent increase/decrease and successive changes ✓ Two-way tables and conditional probability ✓ Mean, median, standard deviation, and outliers ✓ Scatterplots, line of best fit, and residuals ✓ Probability from tables and predicting counts

SAT Strategy for This Section

1. Read the question last — scan the table/graph first to understand the data
2. Identify what the denominators should be — marginal vs. conditional probability
3. Watch for traps: part-to-part vs. part-to-whole ratios
4. Use estimation — if a scatterplot has a clear trend, estimate before calculating

Common Mistakes

• Confusing "percent increase" with "percentage points"
• Using the wrong total for conditional probability
• Forgetting that percent change compounds (not additive)
• Extrapolating beyond the data range when the question asks for interpolation

Deep Dive: Mixed SAT Data Problems

Worked Example 1: Multi-Concept Problem

Worked Example 2: Comprehensive Table + Probability

SAT Problem Solving Cheat Sheet

Time Management for This Section

SAT Problem Solving Challenge 🎯

Problem Solving Quick Check — Select the correct answer.

Full Topic Summary: Problem Solving & Data

Top Strategies

1. Read the question carefully — identify what the denominator should be
2. Use multipliers for percent problems
3. 1 minus for "NOT" probability questions
4. Median when data has outliers
5. Check your answer — does it make sense in context?

🎉 Problem Solving & Data complete!