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Probability

Compute simple and conditional probabilities, usually from a two-way table of counts.

Problem-Solving and Data AnalysisMath
Practice probability free

What the SAT actually asks

You find the chance of an event, almost always as a fraction of counts. Most questions supply a two-way table that splits a group by two characteristics, and ask for the probability that a randomly chosen member has some property. Conditional probability is the main challenge: the question restricts the pool to one row or one column first, which changes the denominator. Some questions give a total and a probability and ask you to work backward to a missing count in the table.

Probability typically accounts for around 1 to 2 of the 44 math questions, inside the roughly 15 percent data share.

Where students lose points

How to attack these questions

  1. 1Find the phrase that restricts the group, usually given that or among, and circle the row or column it points to.
  2. 2Write the fraction as the count that satisfies the condition over the count in the restricted group.
  3. 3Leave the answer as a fraction unless the question asks for a decimal, and check that it is between 0 and 1.

Example questions

Written for this skill, with the answer explained. 92 more are in the app.

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Probability
Real question

A standard deck of 52 playing cards contains 13 cards of each suit. If one card is drawn at random, what is the probability that it is a diamond?

Easier

A fair 12-sided die with faces numbered 1 through 12 is rolled once. What is the probability of rolling a number that is a multiple of 3?

  • A) 5/12
  • B) 1/12
  • C) 4/13
  • D) 1/3 ✓

Why: List the multiples of 3 from 1 to 12: 3, 6, 9, and 12, so 4 faces are favorable. The die has 12 faces, so P = 4/12 = 1/3. 1/12 counts only one favorable face. 5/12 counts one face too many; there are only four multiples of 3 in this range. 4/13 has the right count on top but divides by 13 outcomes when the die has 12.

Medium

Researchers tagged 120 birds by species and by whether they migrated that winter. Migrated Stayed Total Species A 32 8 40 Species B 24 56 80 Total 56 64 120 Given that a bird migrated that winter, what is the probability that it is Species A?

  • A) 4/7 ✓
  • B) 4/15
  • C) 3/7
  • D) 4/5

Why: The bird is known to have migrated, so work inside the migrated column, which holds 56 birds. Of those, 32 are Species A. So the probability is 32/56 = 4/7. 4/15 is 32/120, which divides by the grand total and gives a joint probability instead of a conditional one. 4/5 is 32/40, which divides by the Species A row and answers the reversed question, the chance of migrating given Species A. 3/7 is 24/56, the Species B share of the migrating birds.

Medium

The table shows the results of a quality check on 500 parts. If one of the parts that failed the check is selected at random, what is the probability that it was made on Line 1?

  • A) 0.084
  • B) 0.15556
  • C) 1/2 ✓
  • D) 0.54

Why: The part comes from the 42 + 42 = 84 failed parts, so the denominator is the "failed" column total: 42/84 = 1/2. Choosing 42/270 divides by the Line 1 row total, which answers the reverse question, the probability that a Line 1 part failed. Answering 0.084 means you divided by the grand total instead of the column total. Answering 0.15556 means you divided by the Line 1 row total, reversing the condition. Answering 0.54 means you divided the Line 1 row total by the grand total.

Questions students ask

What is conditional probability on the SAT?

It is the probability of one event given that another is already known to be true. The known condition shrinks the pool, so the denominator becomes the total for that row or column instead of the grand total of the table.

How do I read a two-way table for a probability question?

Identify which characteristic the rows represent and which the columns represent, then locate the single cell that satisfies both parts of the event. Use the row total, the column total, or the grand total as the denominator depending on what the question restricts.

Do I ever need the probability formulas for combinations on the SAT?

No. Digital SAT probability questions are built from counts you can read or compute directly. You need the idea of favorable outcomes over total outcomes and the idea of restricting the pool, not permutation or combination formulas.

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Questions on probability alone, with explanations on every one, and the app tracks which ones you keep missing.

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