Probability & Conditional Probability

Unit 3 · Problem-Solving & Data Analysis

What the SAT asks here

Probability on the SAT is about counting favorable outcomes over total outcomes, reading two-way tables, and handling conditional probabilities (the dreaded 'given that' clause). About one problem per test.

Basic probability

P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}

Complement

P(\text{not }A) = 1 - P(A)

Conditional probability

P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}

Two-way table form

P(A \mid B) = \frac{\text{count in } A \cap B}{\text{total in row/column } B}

SAT Tip: Phrases like 'of those who…' or 'given that…' signal conditional probability. Divide by the restricted subgroup total, not the overall total.

Caution: P(A and B) is not the same as P(A | B). The first uses the overall total; the second uses B's total.

Type 1

Basic probability

Count favorable outcomes and divide by total.

Example 1

A bag contains 5 red, 3 blue, and 2 green marbles. What is the probability of drawing a blue marble at random?

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Type 2

Two-way table

Read counts directly from the table and divide by the appropriate total — overall when the question is unconditional, row/column when it's conditional.

Example 2

Of 100 students surveyed, 35 are girls who like math. What is the probability that a randomly selected student is a girl who likes math?

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Type 3

Conditional probability

Restrict to the subgroup named in the 'given that' clause; divide by that subgroup's total instead of the overall total.

Example 3

Of 35 students who dislike math, 20 are boys. Given that a student dislikes math, what is the probability the student is a boy?

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← Two-Variable Data: Models & Scatterplots Inference from Sample Statistics & Margin of Error →