Probability from Data — SAT Math

Introduction

Probability questions from data show up in the Problem-Solving and Data Analysis domain of the SAT Math section. In this note you will practice three tightly related skills: reading a probability directly from a one-way frequency table or description, converting a frequency into a relative frequency, and—most importantly—working backward from a given probability to find an unknown count. Two-way table and conditional probability questions are covered in the sibling note Conditional Probability and Two-Way Tables; this note stays focused on single-variable data.

Core Concept

Probability as a Fraction

Every basic probability question reduces to one ratio:

P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}

When the data come from a frequency table, the numerator is the count in the category of interest and the denominator is the sum of all counts. Nothing more. The most common error is using the wrong denominator—for example, using only the counts you happen to notice rather than the full total.

Relative frequency is the same ratio applied to observed data instead of theoretical equally-likely outcomes. Conceptually, they are identical on the SAT.

Working Backward: Finding an Unknown Count

Sometimes the probability is given and you must find a missing frequency. Two setups appear:

Case 1 — total is known. Multiply.

\text{unknown count} = P(\text{event}) \times \text{total}

Case 2 — total is also unknown. Let $x$ be the missing count and write an equation.

\frac{x}{(\text{known sum}) + x} = p

Cross-multiply and solve for $x$ .

Quick illustration (Case 2): A box has 45 red and 15 blue balls, plus an unknown number of green balls. $P(\text{green}) = \tfrac{1}{4}$ .

\frac{x}{60+x} = \frac{1}{4} \implies 4x = 60+x \implies 3x = 60 \implies x = 20

Check: $20/80 = 1/4$ . ✓

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Introduction

Core Concept

Probability as a Fraction

Working Backward: Finding an Unknown Count

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