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Probability for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about probability in one place: basic probability, two-way tables, independent and dependent events, the and/or rules, and the complement rule — with step-by-step examples, worked problems, and 2 free practice quizzes.
Practice these quizzes
Free · No signupWhat is probability?
Probability measures how likely an event is to happen, expressed as a number between 0 and 1 (or 0% to 100%). A probability of 0 means an event is impossible; a probability of 1 means it's certain.
PSAT probability questions almost always come from a specific, countable group — a table of survey results, a bag of items, a deck of cards. Identify the total group and the group of interest before doing any calculation.
The basic probability formula
A jar contains 6 red, 4 blue, and 5 green marbles. What is the probability of drawing a green marble?
Two-way tables
| Own a Pet | No Pet | Total | |
|---|---|---|---|
| Freshmen | 32 | 28 | 60 |
| Seniors | 45 | 15 | 60 |
| Total | 77 | 43 | 120 |
Using the table above, what is the probability that a randomly selected senior owns a pet?
When a question restricts to one group ("given that the student is a senior"), the denominator must be that group's total, not the grand total of the whole table.
Independent vs. dependent events
Independent events
The outcome of one event has no effect on the other. Example: rolling a die twice.
Dependent events
The outcome of one event changes the probability of the other. Example: drawing two cards without replacement.
The key phrase to watch for is "without replacement" — it always signals dependent events, since removing an item changes the total for the next draw.
The "and" & "or" rules
| Rule | Formula | When to use |
|---|---|---|
| "And" (independent) | P(A and B) = P(A) · P(B) | Both events 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 together |
A fair coin is flipped twice. What is the probability of getting heads both times?
The simple "or" formula only works when the two events are mutually exclusive. If events can overlap, subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B).
The complement rule
The probability it rains tomorrow is 0.35. What is the probability it does NOT rain?
The complement rule is especially useful for "at least one" questions — it's often much faster to find the probability that none of the desired outcomes occur, then subtract from 1.
Common PSAT traps
- Wrong denominator with a restricted group: 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.
- Forgetting "without replacement" changes the total: the second event's probability uses a smaller total once an item is removed.
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 |
| "At least one" question | Use the complement rule — find P(none), then subtract from 1 |
Now put it to work
Two quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Probability 1
Basic probability and reading two-way tables.
Start quiz → Quiz 2Probability 2
Independent/dependent events, the and/or rules, and the complement rule.
Start quiz →