SAT Math · Problem-Solving and Data Analysis · Skill 5 of 7

Probability and conditional probability

Find probabilities from lists, two-way tables and word problems. Learn to spot the word “given”, pick the right total, and predict how often something should happen.

9 chapters plus a challengeAbout 40 minutesFree, no sign-up
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Hi, I’m Io. Every probability is a fraction. The whole skill is choosing the right number for the bottom.
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One table, three probabilities

Explore · 5 min

This two-way table shows 160 students, sorted by whether they have a part-time job and by grade. A probability is the part you want over the group you are choosing from. Pick the group, then tap a cell, and see how the fraction changes.

Probability basics

Learn · 4 min

P(event) = favourable outcomes ÷ total outcomes
  • A probability is always between 0 and 1. As a percent, between 0% and 100%
  • Not happening: P(not A) = 1 − P(A)
  • The SAT accepts a probability as a fraction or a decimal, so 3/10 and 0.3 are both fine

Reading two-way tables

Practise · 5 min

When a person is chosen from everyone in the table, divide by the grand total in the bottom right corner. The top of the fraction is either a single cell (“and”) or a row or column total (one feature only).

Worked example: 160 students (the table from chapter 1)
Grade 11Grade 12Total
Has a job243660
No job5644100
Total8080160
  1. P(has a job) uses the row total: 60 ÷ 160 = 3/8
  2. P(has a job and is in grade 12) uses one cell: 36 ÷ 160 = 9/40
  3. Both divide by 160, because the student comes from the whole table

Conditional probability

Learn · 5 min

A conditional probability tells you the person comes from one group. Written P(A | B), read “the probability of A given B”. Only group B counts now, so the bottom number is B’s total.

P(A | B) = number in both ÷ number in B
Worked example: a grade 12 student is chosen at random. What is the probability the student has a job?
  1. The group is grade 12, so the bottom number is the grade 12 total: 80
  2. Grade 12 students with a job: 36
  3. P(job | grade 12) = 36 ÷ 80 = 9/20
  4. Compare P(grade 12 | job) = 36 ÷ 60 = 3/5. Swapping the order changes the answer

Watch the orderP(A | B) and P(B | A) share the same top number but have different bottom numbers. The SAT almost always offers the swapped one as a wrong answer.

Which total goes on the bottom?

Sort · 4 min

100 people were asked about their pets. Rows are the pet, columns are adults and children. Before any arithmetic, decide which total each question divides by.

AdultsChildrenTotal
Dog303060
Cat301040
Total6040100

Build the table yourself

Practise · 5 min

Many SAT questions describe the groups in words. Sketch a two-way table, put the overlap in first, then fill in the rest by subtracting.

200 people were surveyed. 120 drink tea, 90 drink coffee, and 50 drink both.
CoffeeNo coffeeTotal
Tea50?120
No tea??80
Total90110200

“Or” and expected counts

Practise · 5 min

  • A or B: add every cell that is A, B or both. Each cell once. The formula version is P(A) + P(B) − P(A and B)
  • Expected count: probability × number of trials. If 15% of bulbs are faulty, expect 0.15 × 400 = 60 faulty bulbs in 400
Grade 11Grade 12Total
Has a job243660
No job5644100
Total8080160

Probability in real life

Scenarios · 6 min

Each tab is a situation you could meet on test day. Answer every question to finish the chapter.

Spot the trap

Detective · 3 min

Each solution has one faulty step. All three slips appear as wrong answer choices on the SAT.

Final challenge

8 questions · 10 min

Key ideas to carry in
ProbabilityPart you want ÷ group you choose from.
Whole tableDivide by the grand total.
GivenDivide by the given group’s total.
Order mattersP(A | B) and P(B | A) are different.
OrAdd the cells that fit, each once.
Expected countProbability × number of trials.
Challenge