Probability & conditional probability
Probability is favorable outcomes over total outcomes. On the SAT, the data almost always comes from a two-way table, and the only real difficulty is choosing the correct total — the whole group, or just one row or column.
What College Board tests
Reading a two-way frequency table to compute a simple probability, a conditional probability (the chance of one category given that we already know another), and probabilities of combined events. The challenge is identifying the right denominator.
The one rule that matters
The word "given" (or "of the students who…", "among the…") is the signal to shrink your denominator from the grand total to a single row or column.
The table these examples use
A school surveys 150 students about their lunch choice. The results:
| Pizza | Salad | Total | |
|---|---|---|---|
| Grade 9 | 45 | 30 | 75 |
| Grade 10 | 35 | 40 | 75 |
| Total | 80 | 70 | 150 |
Four worked examples in SAT format. Read the approach, try it yourself, then tap Show the full solution.
1 · A simple probability
If one of the 150 students is selected at random, what is the probability that the student chose salad?
Approach No "given" here, so the total is everyone — all 150 students. The favorable outcomes are all the students who chose salad, from the salad column's total.
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Answer: A
Why the other choices are wrong
B Uses the pizza total (80) instead of salad.
C Uses 40/70 — a conditional, not this simple probability.
D Uses only the Grade 9 salad cell (30), not the whole salad column.
2 · A conditional probability
Of the students who chose salad, what is the probability that a randomly selected one is in Grade 10?
Approach "Of the students who chose salad" restricts the group: the denominator is the salad total (70), not 150. The favorable outcomes are the Grade 10 students within that salad column.
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Answer: C
Why the other choices are wrong
A Divides the 40 by 150 then mis-reduces.
B Uses the grand total 150, ignoring "of the students who chose salad."
D Uses the salad-over-total simple probability instead.
3 · Conditional, the other direction
Among the Grade 9 students, what is the probability that a randomly selected student chose pizza?
Approach "Among the Grade 9 students" restricts the group to the Grade 9 row, total 75. The favorable outcomes are the Grade 9 students who chose pizza.
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Answer: C
Why the other choices are wrong
A Is 45/150 — uses the grand total, not the Grade 9 row.
B Uses 45/80 (the pizza column total) by mistake.
D The unreduced grand-total version, ignoring "among the Grade 9 students."
4 · A combined event
If one of the 150 students is selected at random, what is the probability that the student is in Grade 9 and chose pizza?
Approach No "given" — this is a simple probability over all 150. The favorable outcomes are a single cell: the students who are both Grade 9 and pizza.
Show the full solution
Answer: B
"And" across both categories points to one cell (45). Because there's no "given," the denominator stays the full 150 — unlike the conditional questions above.
Why the other choices are wrong
A Is the conditional P(pizza given Grade 9) = 45/75, not this combined event.
C Uses 45/80 (pizza column).
D Rounds carelessly to one-half.