Digital SAT · Math · Unit 3 · Problem Solving & Data

Lesson 3.3 · Probability

On the test: 1–2 questions a test, usually read from a two-way table

Big idea. Probability = favourable outcomes ÷ all possible outcomes. The only hard part is deciding what “all” means: when a question says “given that…”, the whole shrinks to just that group, and you count only inside it.

🧑‍🏫 Rule 1 · Count, then dividestep 1 of 5

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📋 The question · SAT · June 2023 · Math 800 Book Q1
A book has 250 pages; 21 have an illustration. If a page is selected at random, what is the probability it has an illustration? (decimal or fraction)
(type the answer)
the question
$250 pages, 21 illustrated$
June 2023: one page chosen at random.

🧑‍🏫 Rule 2 · All the probabilities add to 1step 1 of 3

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📋 The question · SAT · March 2024 · Math 800 Book Q2
At a convention center there are 325 visitors, each in room A, B, or C. The probability of selecting a visitor in room A is 0.48, and in room B is 0.16. How many visitors are in room C?
1. A)25
2. B)36
3. C)104
4. D)117
the question
$P (A) = 0.48, P (B) = 0.16$
March 2024: 325 visitors in rooms A, B or C.

🎛 Move it · shrink the table

August US 2023: probability of Group A GIVEN at least 10 years old. Choose the given columns (blue) and the event (pink).


14	19	27
27	7	26
19	34	7

event:

Blue = the GIVEN part (the new whole) = 120. Pink outline = the event inside it = 46. Probability = 46/120 ≈ 0.383.

Tap header buttons to choose the condition (none selected = everyone).

🧑‍🏫 Rule 3 · “Given that” changes the denominatorstep 1 of 5

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📋 The question · SAT · August US 2023 · Math 800 Book Q9
Group A: 14, 19, 27; Group B: 27, 7, 26; Group C: 19, 34, 7 (ages 0–9, 10–19, 20+; each column total 60). Probability of Group A, given at least 10 years old?
(type the answer)
the question
$P (A ∣\geq 10 yrs)$
August US 2023.

🧑‍🏫 Rule 4 · Percent tables work the same waystep 1 of 4

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📋 The question · SAT · May 2024 · Math 800 Book Q5
Ages in a city: under 18: 27%; 18–40: 24%; 41–65: 28%; over 65: 21%. Closest probability a person is over 65, given at least 18?
1. A)0.21
2. B)0.29
3. C)0.35
4. D)0.41
the question
$P (> 65 ∣\geq 18)$
May 2024: under 18: 27%, 18–40: 24%, 41–65: 28%, over 65: 21%.

Fast lane · read the question backwards

The words after “given that” name the DENOMINATOR row/column. The words before name what you count inside it.

Real question 1SAT · August 2024 · Math 800 Book Q10type the answer

A bin has T-shirts: Flames — small 6, medium 24, large 18 (total 48); Panthers — small 9, medium 33, large 24 (total 66). What is the probability of selecting a medium T-shirt, given that it is a Flames T-shirt?

🧑‍🏫 Rule 5 · NOT: the complementstep 1 of 4

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the idea
$P (not A) = 1 - P (A)$
Every outcome either is A or is not A, and together they make the whole (probability 1).

🧑‍🏫 Rule 6 · OR: add, then remove the overlapstep 1 of 5

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the rule
$P (A or B) = P (A) + P (B) - P (A and B)$
Adding counts the people who are in BOTH groups twice, so subtract them once.

🧑‍🏫 Rule 7 · AND: multiply (one thing after another)step 1 of 4

1
independent
$P (A and B) = P (A) \times P (B)$
When the first result does not change the second — two separate coin flips, two dice.

Why it works · why multiply for AND

If 8 of 20 first picks are red, and then 7 of 19 second picks are red, the successful pairs are 8 × 7 out of 20 × 19 possible pairs.

🧑‍🏫 Rule 8 · Expected count = probability × how manystep 1 of 3

1
the idea
$expected = P \times N$
If a result happens with probability $P$ , then out of $N$ tries we expect about $P \cdot N$ .

🧑‍🏫 Rule 9 · Row percent vs column percentstep 1 of 5

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📋 The question · SAT · November 2024 · Math 800 Book Q15
List A: 12 even, 44 odd; List B: 40 even, 4 odd. What fraction of the even integers are in list A?
1. A) $\frac{12}{52}$
2. B) $\frac{12}{56}$
3. C) $\frac{52}{100}$
4. D) $\frac{52}{56}$
the question type
$“what fraction of the EVEN integers are in list A?”$
November 2024.

🧑‍🏫 Rule 10 · Probability is always between 0 and 1step 1 of 3

1
range
$0 \leq P \leq 1$
0 = impossible, 1 = certain.

Desmos move · calculator

The Bluebook calculator keeps fractions exact: type 46/120 and it shows 0.3833. For typed answers, enter the fraction itself.

Take with you

▸P = favourable ÷ total; answer as a fraction or decimal between 0 and 1.
▸The probabilities of all the separate parts add to 1; P × total = expected count.
▸“Given that …” / “of the …” → that group is the new total; count only inside it.
▸NOT: $1 - P (A)$ . “At least one” = $1 - P (none)$ .
▸OR: add, then subtract the overlap (“both”).
▸AND: multiply; without replacement, the second fraction loses one from top and bottom.

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