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Probability & Conditional Probability for the SAT: Complete Study Guide + Free Practice Problems
Everything the SAT tests about probability in one place: basic probability, two-way tables, conditional probability, independent vs. dependent events, and the and/or rules — with step-by-step examples, worked problems, and a free practice quiz.
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Free · No signupProbability & Conditional Probability
Basic probability, two-way tables, conditional probability, and independent/dependent events.
What is probability?
Probability measures how likely an event is to happen, expressed as a number between 0 and 1 (or equivalently, 0% to 100%). A probability of 0 means an event is impossible; a probability of 1 means it's certain.
The SAT's probability questions almost always come from a specific, countable group — a table of survey results, a bag of marbles, a deck of cards. Identify the total group and the group of interest before doing any calculation.
The basic probability formula
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of drawing a blue marble?
Two-way tables
A two-way table organizes data by two categories at once — for example, gender and favorite subject. Most SAT conditional probability questions are built directly from a two-way table.
| Prefers Math | Prefers Science | Total | |
|---|---|---|---|
| Freshmen | 40 | 35 | 75 |
| Sophomores | 30 | 45 | 75 |
| Total | 70 | 80 | 150 |
Using the table above, what is the probability that a randomly selected student is a freshman?
Conditional probability
Conditional probability asks for the probability of an event given that another condition is already known to be true. The key move is to shrink your total from the whole group down to only the group that satisfies the condition.
Using the table above, what is the probability a randomly selected student prefers Math, given that the student is a sophomore?
The most common conditional probability error is dividing by the grand total instead of the restricted total. Once you see the word "given," your denominator must become the size of the given group only — never the full table total.
Independent vs. dependent events
Independent events
The outcome of one event has no effect on the other. Example: flipping a coin twice.
Dependent events
The outcome of one event changes the probability of the other. Example: drawing two cards without replacement.
A fast test for independence using a two-way table: if P(A given B) equals the plain P(A), the events are independent. If those two values differ, the events are dependent.
The "and" & "or" rules
| Rule | Formula | When to use |
|---|---|---|
| "And" (independent) | P(A and B) = P(A) · P(B) | Both events must happen, and they don't affect each other |
| "Or" (mutually exclusive) | P(A or B) = P(A) + P(B) | Either event happens, and they can't happen at the same time |
A fair coin is flipped, and a fair six-sided die is rolled. What is the probability of getting heads AND rolling a 4?
The simple "or" formula (adding probabilities) only works when the two events are mutually exclusive — they cannot both happen at once. If the events can overlap, you must subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B).
Common SAT traps
- Wrong denominator in conditional probability: always shrink the total to the given condition, not the full data set.
- Assuming independence without checking: confirm two events are actually independent before multiplying their probabilities.
- Double-counting overlap in "or" problems: subtract the "and" probability when events aren't mutually exclusive.
- Misreading two-way table row/column totals: double-check whether a question is asking about a row total, a column total, or the grand total.
Test day strategy for probability
| Question signal | Fastest approach |
|---|---|
| Basic probability from a group | Favorable outcomes over total outcomes |
| "Given that ___" language | Restrict your denominator to only the given condition |
| Two independent events, both must happen | Multiply the two individual probabilities |
| Either of two mutually exclusive events | Add the two individual probabilities |
| Events that could overlap | Add the probabilities, then subtract the overlap |
| Two-way table given | Identify exactly which row, column, or cell the question refers to before dividing |
Now put it to work
One comprehensive quiz covering the full topic, from basic probability through conditional probability and independent events.
Probability & Conditional Probability
Basic probability, two-way tables, conditional probability, and independent/dependent events.