Math

Probability

5 min readEasy5-question drill

SAT probability problems are mostly about reading two-way tables correctly and remembering that probability = (favorable outcomes) / (total outcomes). Two formulas, one careful read.

Where this lesson fits

Before

Statistics→Ratios and Proportions→

Probability

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→Data Interpretation

The basic formula:

$P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{number of total outcomes}}$

Probabilities are always between 0 and 1. $P = 0$ means impossible; $P = 1$ means certain.

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Probability is always favorable / total

Scenario	Numerator	Denominator	Probability
Bag has 4 red, 6 blue, 5 green; pick one	4 (red)	15 (total)	$4/15$
Same bag; pick one, not blue	9 (red + green)	15 (total)	$9/15$
From a class of 30, 18 are girls; pick a girl	18	30	$18/30$
From the 18 girls, 12 wear glasses; pick a glass-wearer	12	30 OR 18	depends on framing

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?" → $\frac{65}{80}$ (whole table is the population).
"Given that a student is a girl, what's the probability she passes?" → $\frac{35}{40}$ (only girls is the population).
"Given that a student passed, what's the probability she's a girl?" → $\frac{35}{65}$ (only passers is the population).

This is conditional probability — given X, what's the probability of Y — and the SAT loves it.

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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*

The 'and' / 'or' rules. Less commonly tested but worth knowing:

$P(A \text{ and } B) = P(A) \cdot P(B)$ if events are independent.
$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ .

Complement rule. $P(\text{not } A) = 1 - P(A)$ . Useful when computing not at all probabilities.

"Probability that at least one of three students passes." → easier as $1 - P(\text{all three fail})$ .

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.

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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 survey classified respondents into four groups: Group A (24), Group B (36), Group C (28), Group D (36). If one respondent is chosen at random, what is the probability they are in Group A?

9/31

2/5

6/31

7/31

Worked examples

Example 1

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?

Example 2

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?

Common pitfalls

Using the wrong denominator

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).

Treating non-independent events as independent

Drawing two cards without replacement → the second draw's probability depends on the first. $P(\text{both red}) \neq P(\text{red}) \cdot P(\text{red})$ — the count changes after the first draw.

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Forgetting to subtract overlap in 'or' questions

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ . If you don't subtract the overlap, you double-count students who fall into both categories.

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Confusing 'at least one' with the simple probability

At least one of 3 students passes is NOT the same as $3 \cdot P(\text{pass})$ . Use the complement: $P(\text{at least one}) = 1 - P(\text{none})$ .

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Key takeaways

$P =$ favorable outcomes / total outcomes. Always between 0 and 1.
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The denominator is the given group: given X, the population is X-only.
$P(\text{not } A) = 1 - P(A)$ . Use this when the question is about not, neither, or at least one.
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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.

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Try it yourself

5 practice questions on Probability, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Before

Statistics→Ratios and Proportions→

Probability

You are here

→Data Interpretation