This unit is ≈8% of the SAT, across 6 lessons. Full syllabus
Lesson 5 of 6 · Problem-solving and data analysis
Probability and conditional probability
7 min read · about 1 h with practice3 quick checks<1% of the testCore: Core: tested on most papers
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SAT probability questions are short, and most are built around one trap: dividing by the wrong group. The majority give you a two-way table (or a graph of counts) and ask for a simple or conditional probability. The rest ask for an expected count or use a little algebra. Expect about one or two per test in Problem-Solving and Data Analysis, as multiple choice or student-produced response (SPR). They are some of the most reliable points on the Math section once you have a method.
By the end you’ll be able to
Compute probabilities and relative frequencies from counts and two-way tables
Compute conditional probabilities by restricting to the given group (the row or column named after ‘given’)
Use complements and basic independence reasoning
What the exam asks
Nearly every probability item fits one of these patterns.
Pattern
Typical wording
What to compute
Simple probability
“If one of the 240 shoppers is selected at random, what is the probability that...”
part ÷ grand total
Conditional probability
“If a shopper is selected at random from those who bought fruit...” or “given that...”
part ÷ total of the named group
Expected count
“Based on the table, how many of 1,500 seeds would be expected to...”
rate from the matching group × number of new items
Missing information
a table with blank cells, or percents in place of counts
fill in what you need first
Algebraic probability
“After 10 red chips are added, the probability becomes 21...”
vii.Check your understanding
3 questions on probability and conditional probability. Every option is explained once you answer.
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PromptCard 1 of 3
Probability of an event when all outcomes are equally likely
write an equation in the unknown count
Multiple-choice answers are often fractions, or decimals introduced by “closest to”. In SPR items you can type a fraction such as 7/30 directly.
Core ideas
Probability is a fraction of a group
When every outcome is equally likely,
P(event)=total number of outcomesnumber of favorable outcomes
A probability is always between 0 and 1. If your answer is greater than 1, the fraction is upside down.
Reading a two-way table
A two-way table sorts one group by two categories. Here is a table for 200 gym members, sorted by the time they usually exercise and whether they use weights.
Morning
Evening
Total
Uses weights
36
64
100
Does not use weights
44
56
100
Total
80
120
200
There are three kinds of probability:
Kind
Example
Calculation
Joint (both conditions)
P(morning and uses weights)
20036=0.18
Marginal (one condition)
P(morning)
20080=0.40
Conditional (restrict first)
P(uses weights, given morning)
8036=0.45
Some tables show percentages of all respondents instead of counts. Treat those percents exactly like counts out of 100. A cell percent is a joint probability, and a conditional probability is a cell percent divided by the given group’s total percent.
Conditional probability: shrink the denominator
P(A∣B)=number in Bnumber in both A and B
The group named after “given that”, “from those who”, “of the” or “among” is B. Its total is your denominator. The numerator is the part of that group that has the feature you want.
Order matters
P(weights∣morning)=8036=0.45, but P(morning∣weights)=10036=0.36. The numerator is the same and the denominator is different. The SAT routinely puts both in the answer choices.
Complements
P(not A)=1−P(A). In a table, “not” means adding the other cells in the group, or subtracting from the group total. When every item falls into exactly one of several categories, the probabilities add to 1, so a missing category’s probability is 1 minus the others.
Expected counts
If a fraction p of a group has a feature, the expected number among N similar new items is pN. The key word is similar: use the rate from the row or column that matches the new items, not the overall rate.
Independence and “percent of a percent”
Events A and B are independent when knowing one doesn’t change the chance of the other: P(A∣B)=P(A). Then P(A and B)=P(A)⋅P(B). In the gym table, P(weights)=0.50 but P(weights∣morning)=0.45, so the two aren’t independent. The SAT more often chains percents: “60% of students take a world language; of those, 35% take Spanish” means P(Spanish)=0.60×0.35=0.21. A percent of a percent is found by multiplying, never by adding.
Worked examples
Exam technique
Name the group before you calculate. Find the words after “given”, “from those who”, “of the” or “among” and write that total as the denominator before looking for the numerator.
Stay inside the group for the numerator. It is one cell, or the sum of the cells in that row or column that match.
No condition means the grand total. “If one of the 500 subscribers is selected at random...” with nothing else restricts nothing.
Combine rows or columns when the group spans them. “30 years or older” can mean adding two age rows before dividing.
Use the decimal for “closest to” choices. Divide in Desmos or by hand. The wrong choices are usually the correct numerator over the wrong total.
SPR entry. Type the fraction (for example, 9/22); it doesn’t have to be reduced. If you type a decimal, fill every space: .4090, .4091 or 0.409.
Budget. A table item should take 45–75 seconds. Spend the time you save on the harder module 2 questions.
Common mistakes
Quick recap
Probability = favorable ÷ total, always between 0 and 1.
Joint: cell ÷ grand total. Marginal: row or column total ÷ grand total. Conditional: cell ÷ total of the given group.
“Given”, “from those who”, “of the” and “among” restrict the denominator.
P(A∣B)=P(B∣A) in general, and both appear as choices.
Complement: P(not A)=1−P(A).
Expected count = rate from the matching group × number of new items.
Fill in missing cells with the row and column totals, and only the cells you need.
For unknown counts, write the probability as a fraction, set it equal to the given value and solve (Desmos can do this).