SAT Math – Problem Solving & Data Analysis
Probability and Relative Frequency
Calculating likelihood and interpreting experimental data
Probability and relative frequency represent two approaches to measuring likelihood—theoretical prediction and experimental observation. On the SAT, you'll calculate probabilities, interpret survey data, use two-way tables, and understand conditional probability—skills that form the foundation of statistical reasoning and data-driven decision making.
Success requires understanding basic probability principles, calculating probabilities from data, distinguishing independent and dependent events, using relative frequency to estimate probability, and interpreting conditional statements. These aren't abstract concepts—they're the tools used in medicine (risk assessment), insurance (premium calculation), sports analytics, weather forecasting, and countless everyday decisions involving uncertainty.
Understanding Probability and Relative Frequency
Theoretical Probability
Theoretical probability is the likelihood of an event calculated from mathematical reasoning, assuming all outcomes are equally likely.
Range: 0 ≤ P ≤ 1 (or 0% to 100%)
Example: Probability of rolling a 3 on a die = \(\frac{1}{6}\)
Relative Frequency (Experimental Probability)
Relative frequency is the probability estimated from actual experimental or observational data.
Purpose: Estimate probability from real data
Example: If heads appeared 52 times in 100 coin flips, relative frequency = \(\frac{52}{100} = 0.52\)
Conditional Probability
Conditional probability is the probability of an event given that another event has already occurred.
Formula: \(P(A | B) = \frac{\text{Number with both A and B}}{\text{Number with B}}\)
Example: Probability a student has a pet given they own a dog
Complementary Events
Complementary events are outcomes that together account for all possibilities. One happens if and only if the other doesn't.
Key idea: Sum of complementary probabilities = 1
Example: If P(rain) = 0.3, then P(no rain) = 0.7
Essential Formulas & Concepts
Basic Probability Formula
\(P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}\)
Probability must be between 0 and 1 (inclusive)
Can be expressed as fraction, decimal, or percentage
Example: \(\frac{3}{10} = 0.3 = 30\%\)
Compound Probability (Independent Events)
For independent events (one doesn't affect the other):
\(P(A \text{ and } B) = P(A) \times P(B)\)
Example: Flipping heads twice = \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)
Conditional Probability from Two-Way Tables
\(P(A | B) = \frac{\text{count in both A and B}}{\text{count in B}}\)
Find the intersection cell, divide by the row or column total
Probability Properties
• Impossible event: P = 0
• Certain event: P = 1
• Sum of all possible outcomes = 1
• \(P(A) + P(\text{not } A) = 1\)
Common Pitfalls & Expert Tips
❌ Using wrong denominator for conditional probability
For P(A | B), divide by the count of B, not the total! You're restricting to the subset where B already happened.
❌ Confusing "and" with "or" in probability
"And" typically means multiply (for independent events). "Or" means add (for mutually exclusive events). Know which operation applies!
❌ Forgetting probabilities must sum to 1
All possible outcomes together have probability 1. Use this to check your work or find missing probabilities.
❌ Mixing up rows and columns in tables
Carefully identify which variable is in rows vs. columns. Read labels before calculating!
✓ Expert Tip: Use complements for "at least one"
For "at least one" problems, calculate P(none) and subtract from 1. Much faster than adding all individual cases!
✓ Expert Tip: Organize with tree diagrams or lists
For compound events, list or draw all possible outcomes systematically. This prevents missing cases.
✓ Expert Tip: Read conditional probability carefully
P(A | B) is very different from P(B | A). The event after "|" is what you know has happened—that's your new total.
Fully Worked SAT-Style Examples
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble is selected at random, what is the probability it is blue?
Solution:
Step 1: Count favorable outcomes
Blue marbles = 3
Step 2: Count total outcomes
Total marbles = 5 + 3 + 2 = 10
Step 3: Apply probability formula
\(P(\text{blue}) = \frac{3}{10} = 0.3 = 30\%\)
Answer: \(\frac{3}{10}\) or 0.3 or 30%
A survey of 200 students found that 75 prefer pizza for lunch. Based on this data, what is the relative frequency of students who prefer pizza?
Solution:
Step 1: Identify the components
Students who prefer pizza = 75
Total students surveyed = 200
Step 2: Calculate relative frequency
\(\text{Relative Frequency} = \frac{75}{200} = 0.375 = 37.5\%\)
Interpretation:
This relative frequency can be used to estimate the probability
In a larger population, we'd expect about 37.5% to prefer pizza
Answer: 0.375 or 37.5%
The probability of rain tomorrow is 0.35. What is the probability it will NOT rain tomorrow?
Solution:
Step 1: Identify complementary events
"Rain" and "not rain" are complements
Together they cover all possibilities
Step 2: Use complement formula
\(P(\text{not rain}) = 1 - P(\text{rain})\)
\(= 1 - 0.35 = 0.65\)
Answer: 0.65 or 65%
A fair coin is flipped twice. What is the probability of getting heads both times?
Solution:
Step 1: Find probability of each event
P(heads on first flip) = \(\frac{1}{2}\)
P(heads on second flip) = \(\frac{1}{2}\)
Step 2: Recognize independence
The second flip doesn't depend on the first
For independent events, multiply probabilities
Step 3: Calculate
\(P(\text{heads and heads}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)
All Possible Outcomes:
HH, HT, TH, TT (4 equally likely outcomes)
Only 1 out of 4 is HH, confirming \(\frac{1}{4}\)
Answer: \(\frac{1}{4}\) or 0.25 or 25%
The table shows 100 students by grade and whether they participate in band.
| In Band | Not in Band | Total | |
|---|---|---|---|
| 10th Grade | 15 | 35 | 50 |
| 11th Grade | 20 | 30 | 50 |
| Total | 35 | 65 | 100 |
What is the probability a student is in band GIVEN they are in 11th grade?
Solution:
Step 1: Understand the conditional
We're GIVEN 11th grade, so we only consider that row
P(in band | 11th grade)
Step 2: Find relevant values
11th graders in band = 20
Total 11th graders = 50
Step 3: Calculate conditional probability
\(P(\text{band} | \text{11th}) = \frac{20}{50} = \frac{2}{5} = 0.4\)
Common Error:
Don't use 100 as denominator!
Conditional probability restricts to the given subset (11th grade = 50)
Answer: \(\frac{2}{5}\) or 0.4 or 40%
In a sample of 50 light bulbs, 3 were defective. Based on this relative frequency, about how many defective bulbs would you expect in a shipment of 1,000 bulbs?
Solution:
Step 1: Calculate relative frequency
\(\text{Relative Frequency} = \frac{3}{50} = 0.06 = 6\%\)
Step 2: Apply to larger population
Expected defective = 6% of 1,000
\(0.06 \times 1{,}000 = 60\) defective bulbs
Alternative Method:
Set up proportion: \(\frac{3}{50} = \frac{x}{1000}\)
Cross-multiply: \(50x = 3000\)
\(x = 60\)
Answer: 60 defective bulbs
A basketball player makes free throws with 80% accuracy. If she takes 2 free throws, what is the probability she makes at least one?
Solution:
Step 1: Use complement strategy
"At least one" = 1 − P(none)
Easier to find P(misses both)
Step 2: Find P(misses both shots)
P(miss) = 1 − 0.80 = 0.20
\(P(\text{miss both}) = 0.20 \times 0.20 = 0.04\)
Step 3: Calculate complement
\(P(\text{at least one}) = 1 - 0.04 = 0.96\)
Why This is Easier:
"At least one" includes: makes 1st only, makes 2nd only, makes both
That's 3 cases to add! Complement method = 1 calculation
Answer: 0.96 or 96%
Using the table from Example 5, what is the probability a randomly selected student is in 10th grade?
Solution:
Step 1: Identify relevant values
This is NOT conditional—we're looking at all students
10th graders = 50 (from row total)
Total students = 100
Step 2: Calculate probability
\(P(\text{10th grade}) = \frac{50}{100} = 0.5\)
Compare to Conditional:
P(10th grade) = \(\frac{50}{100}\) uses grand total
P(band | 10th) = \(\frac{15}{50}\) uses row total
Very different questions and denominators!
Answer: 0.5 or 50%
Probability Quick Reference
SAT Probability Checklist
Basic Probability
- Count favorable outcomes carefully
- Count total possible outcomes
- Simplify fractions
- Check 0 ≤ P ≤ 1
Conditional Probability
- Event after "|" is what you know
- Use that as your new total
- Find intersection in two-way table
- P(A|B) ≠ P(B|A)
Compound Events
- Independent? Multiply probabilities
- List all outcomes systematically
- Use complements for "at least"
- Draw tree diagram if needed
Common Traps
- Wrong denominator in conditional
- Adding when you should multiply
- Forgetting complement rule
- Misreading table labels
Probability: Quantifying Uncertainty
Probability is the mathematical language we use to reason about uncertainty and make decisions when outcomes aren't guaranteed. Every time a weather forecaster predicts rain, a doctor assesses treatment risks, a sports analyst calculates win chances, or an insurance company sets premiums, they're applying probability principles. The SAT tests these skills because they represent essential quantitative reasoning for modern life—understanding that a 60% chance of rain doesn't mean it will definitely rain, that conditional probabilities change when you have additional information, and that rare events can still happen. Relative frequency connects theory to reality, showing how we use observed data to estimate probabilities when mathematical calculation isn't possible. Master both theoretical probability and experimental estimation not just for test success, but to become someone who can think critically about risk, interpret statistics correctly, understand medical test results, and make informed decisions in an uncertain world. When you understand probability, you understand that while we can't predict individual outcomes, we can quantify likelihood—and that's often enough to make better choices.
