Math · Problem-Solving and Data Analysis

Probability and Conditional Probability on the Digital SAT

Most probability questions on the SAT hand you counts, usually from a survey or a two-way breakdown described in the question, and ask you to turn those counts into a fraction: the right count on top, the right count on the bottom. The harder ones add a second step, like multiplying two chances together or finding the chance of "none" and subtracting from 1, but even those come down to keeping your counts straight.

Here is the honest part: this skill shows up in low volume, usually one or two questions per test. But those questions are nearly pure logic, and students who have practiced the format pick up the points in under a minute. Students who have never seen it lose easy points, not because the math is hard, but because they have never had to decide which group the question is actually asking about.

The one idea that separates a right answer from a wrong one is conditional probability: when a question says "of the students who play a sport" or "given that the person owns a car," your denominator is that smaller group, not the whole survey. Once that clicks, most of these questions feel almost unfair in how quickly they fall.

New to the Digital SAT? Start with how the test is structured and scored.

What the SAT actually tests

Finding the probability of an event as (favorable outcomes) divided by (total outcomes) from counts described in the question.
Reading two-way data presented in prose or a table: totals for each category, and counts for combinations like "juniors who play a sport."
Conditional probability: when the question restricts to a group ("of the seniors," "given that the student plays a sport"), the denominator becomes that group's total, not the overall total.
Deriving a missing count before computing: the question may give you a total and one piece, and you have to subtract to find the cell you need.
Recognizing complements: the probability a student does NOT have a job is 1 minus the probability they do.
Working with multi-step probability: multiplying independent chances (all three parts pass, both trials succeed), sequential draws without replacement where the second chance depends on the first, and the "at least one" move where you find the chance of none and subtract from 1.
Expressing answers as fractions, decimals, or percents, and simplifying correctly.

Strategies that work

Restrict the denominator first

Before touching any numbers, find the phrase that defines the group being selected from: "a surveyed student," "a junior chosen at random," "a person who plays a sport." That group's total is your denominator. Everything else in the problem is either your numerator or a distraction.

Sketch a quick two-way grid

When the data is described in prose, jot a tiny 2-by-2 grid: rows for one category, columns for the other. Fill in what the question gives you, then subtract to fill in what it does not. Thirty seconds of setup prevents the most common wrong answers.

Say the fraction in words

Translate your answer back into English before bubbling: "40 sport-playing juniors out of 70 total sport players." If the sentence does not match the question exactly, your denominator is probably wrong. This catch takes five seconds and saves real points.

Check that your answer is possible

A probability must be between 0 and 1, and a conditional probability uses a numerator that fits inside its denominator. If your numerator is bigger than your denominator, or the number in a subgroup exceeds the group total, you mixed up two counts somewhere.

Mistakes to avoid

Using the whole survey total as the denominator when the question restricts to a smaller group. "Of the students who play a sport" means the denominator is sport players only.
Flipping the condition: computing P(plays a sport, given junior) when the question asks P(junior, given plays a sport). These are different fractions with different denominators.
Forgetting to derive missing counts. If 110 people own a bike and 80 of them also own a car, the bike-but-no-car count is 30, and the question usually hinges on that subtraction.
Adding overlapping counts twice, for example adding "juniors" and "sport players" without accounting for juniors who play a sport.
Not simplifying, then picking a distractor that looks like the unsimplified fraction of a different pair of counts.

Practice with Nova, your Math coach

Answer right here and Nova walks you through it, just like in a drill. Difficulty labels are relative within this skill.

Tip: the Calculator button on each question opens the same Desmos graphing calculator you get on the real Digital SAT. Try solving these with it.

Question 1Easy

In a survey of 150 students at a high school, 60 students said they usually get to school by bus.

If one of the surveyed students is selected at random, what is the probability that the student usually gets to school by bus?

Pick an answer to get instant coaching from Nova.

Question 2Medium

The table summarizes the results of a survey of 120 students at a school, by class year and whether the student plays a school sport.

Surveyed Students, by Class Year and Sport Participation
	Plays a school sport	Does not play	Total
Juniors	40	30	70
Seniors	30	20	50
Total	70	50	120

If one of the surveyed students who plays a school sport is selected at random, what is the probability that the student is a junior?

Pick an answer to get instant coaching from Nova.

Question 3Hard

In a survey of 200 residents of a town, 120 residents own a car. Of the residents who own a car, 80 also own a bicycle. In total, 110 of the surveyed residents own a bicycle.

If one of the surveyed residents who does NOT own a car is selected at random, what is the probability that the resident owns a bicycle?

Pick an answer to get instant coaching from Nova.

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The rest of Problem-Solving and Data Analysis

Probability and Conditional Probability is one of 7 skills in Problem-Solving and Data Analysis. The ideas build on each other, so it pays to learn them as a set rather than in isolation. See how Problem-Solving and Data Analysis fits together.

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