Solving Probability Questions on the SAT

The method: Table → Favorable → Divide

Every SAT probability question reduces to the same fraction: P(event) = favorable outcomes / total outcomes. A bag of 20 marbles with 8 red ones gives P(red) = 8/20 = 2/5. The only extra work on most SAT questions is figuring out which numbers count as “favorable” and which count as “total” — and a two-way table organizes exactly those counts for you.

A survey of 250 students recorded whether each plays a sport and whether each plays an instrument:

Row and column totals are already favorable/total pairs waiting to be divided. Walk through pulling a conditional probability out of this table below.

Two-way tables and reading the right cell

A two-way table has three kinds of numbers: row totals and column totals (each the total for one category, ignoring the other), single cells (the count for both categories together), and the grand total (everyone). Before dividing anything, decide which of these four numbers the question is actually asking for.

Conditional probability: P(A | B)

P(A | B) means “the probability of A, given that B already happened.” In a two-way table, that means you divide the joint cell by the row or column total for B — never by the grand total. Restrict your attention to just B's row (or column) first, then find A's share of it.

The order matters: P(instrument | sport) and P(sport | instrument) use the same joint cell but different denominators, and they are generally not equal.

Compound events: and / or

For two independent events (a coin flip and a die roll — neither affects the other), multiply: P(A and B) = P(A) × P(B). For mutually exclusive “or” events (a single die roll can't show both a 2 and a 5), add: P(A or B) = P(A) + P(B). If the events can overlap, subtract the overlap once: P(A or B) = P(A) + P(B) − P(A and B).

The most common error is using the wrong operation — adding independent events instead of multiplying them, or forgetting to subtract the overlap for “or” events that can happen together.

Practice until it's automatic

These mix simple probability, table reading, conditional probability, and compound events — the full toolkit for SAT probability questions.

Once table-reading and conditional probability feel automatic, the same “restrict, then divide” instinct applies directly to sample-to-population estimates, where you scale a sample proportion up to a full population.

Practice questions

Here are five practice problems that help you hone your skills. Use the same method we learned earlier above to solve these problems. Remember, practice makes perfect.

Questions 1–3 use this table: a survey of 300 shoppers found that of the 180 who bought a jacket, 117 also bought gloves and 63 did not. Of the 120 who did not buy a jacket, 21 bought gloves and 99 did not.