Two-way tables & probability · SAT Maths

Concept

A two-way table cross-classifies a set of items along two categorical variables. Three kinds of probabilities to read off:

Joint — P(A and B) = (cell count) / (grand total)
Marginal — P(A) = (row or column total) / (grand total)
Conditional — P(A | B) = (cell count) / (B's row or column total)

Conditional probability is the SAT's favourite — the word given tells you to restrict to the corresponding row or column total instead of the grand total.

Two events are independent if P(A and B) = P(A) · P(B).

Worked example 1

A class of 60 students includes 35 who take Math, 25 who take Science, and 10 who take both. What is the probability that a randomly chosen student takes only Math?

Solution

Math only. 35 − 10 = 25 students take Math but not Science.

Probability. P = 25 / 60 = 5/12

P(Math only) = 5/12 ≈ 0.417.

Worked example 2

Same class. Given that a student takes Math, what is the probability they also take Science?

Solution

Condition. Restrict to the 35 Math students.

Both. Of those, 10 also take Science.

Compute. P(Sci | Math) = 10 / 35 = 2/7

P(Sci | Math) = 2/7 ≈ 0.286.

Practice test

8 questions on reading two-way tables, joint vs marginal vs conditional probability, and independence.

Practice test · 8 questions Question 1 of 8 · Score 0