SAT probability problems are mostly about reading two-way tables correctly and remembering that probability = (favorable outcomes) / (total outcomes). Two formulas, one careful read.
From the 18 girls, 12 wear glasses; pick a glass-wearer
12
30 OR 18
depends on framing
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What's the denominator on this two-way-table question?
Does the question say *given that...* or *of those who...*?
Yes ↓
Identify the given group. The denominator is the count in that group only.
Yes ↓
→ Use the GIVEN GROUP'S total as the denominator (not the whole table)
No ↓
→ Re-read — make sure you spotted the given condition
No ↓
Is it asking about a randomly selected person from the whole population?
Yes ↓
→ Denominator = the WHOLE TABLE'S total
No ↓
→ Look for the implicit condition — the question may name a subgroup like *among seniors*
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Quick check
Read carefully. *Given X* signals conditional probability — the denominator is the count in the X group, not the whole table.
A frequency table shows that in a class of 30 students, 12 chose pizza, 8 chose tacos, 6 chose salad, and 4 chose burgers as their favorite lunch. What fraction of the class chose either pizza or tacos?
Worked examples
Example 1
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Example 2
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Common pitfalls
Using the wrong denominator
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Treating non-independent events as independent
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Forgetting to subtract overlap in 'or' questions
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Confusing 'at least one' with the simple probability
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Key takeaways
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Tracks your progress across lessons.
Skill check
10 questions on Probability and Conditional Probability, from our question bank. Score 70% or better across 10 answers to reach Silver and pass this step.
P(event)=number of total outcomesnumber of favorable outcomes
Probabilities are always between 0 and 1. P=0 means impossible; P=1 means certain.
Reading two-way tables. Most SAT probability problems give you a table like:
Pass
Fail
Total
Boys
30
10
40
Girls
35
5
40
Total
65
15
80
The key skill: identify what the denominator is. Different questions use different denominators:
"What is the probability that a randomly selected student passes?" → 8065 (whole table is the population).
"Given that a student is a girl, what's the probability she passes?" → 4035 (only girls is the population).
"Given that a student passed, what's the probability she's a girl?" → 6535 (only passers is the population).
This is conditional probability — given X, what's the probability of Y — and the SAT loves it.
The 'and' / 'or' rules. Less commonly tested but worth knowing:
P(A and B)=P(A)⋅P(B) if events are independent.
P(A or B)=P(A)+P(B)−P(A and B).
Complement rule.P(not A)=1−P(A). Useful when computing not at all probabilities.
"Probability that at least one of three students passes." → easier as 1−P(all three fail).
Common SAT trick: Q phrased as "of those who passed, what fraction were girls?" — this isn't really probability language but the calculation is the same: count the girls who passed, divide by all who passed.
Tip: circle the given condition in the question. That tells you the denominator. Underline what's being asked. That tells you the numerator.
A bag contains 4 red marbles, 6 blue marbles, and 5 green marbles. If one marble is drawn at random, what is the probability that it is not blue?
A school survey of 200 students reports:
Cafeteria
Pack lunch
Total
Freshman
40
20
60
Senior
60
80
140
Total
100
100
200
Given that a student packs lunch, what is the probability the student is a senior?
When a question says given that the student is female, your denominator is all females, not the whole table. Many SAT trap answers use the right numerator but the wrong denominator (usually the table total).
Drawing two cards without replacement → the second draw's probability depends on the first. P(both red)=P(red)⋅P(red) — the count changes after the first draw.
P(A or B)=P(A)+P(B)−P(A and B). If you don't subtract the overlap, you double-count students who fall into both categories.
At least one of 3 students passes is NOT the same as 3⋅P(pass). Use the complement: P(at least one)=1−P(none).
P= favorable outcomes / total outcomes. Always between 0 and 1.
The denominator is the given group: given X, the population is X-only.
P(not A)=1−P(A). Use this when the question is about not, neither, or at least one.
Two-way tables are the SAT's favorite probability format — always identify what subset is the denominator.
Independent events: multiply probabilities for and. Overlapping events: subtract the overlap when using or.