IB Mathematics · Grade 9 · Self-Study

Probability & Set Theory

20 core questions · Multiple choice · Instant feedback

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01 — Set Theory & Operations
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Quick Memory Key

UNION = OR = more → A ∪ B everything in A or B
INTERSECTION = AND = both → A ∩ B only what's in A and B
COMPLEMENT = NOT → A' everything outside A
SUBSET ⊆ = fits inside → every element of A is also in B

Q 01
SetsEasy

Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}.
What is A ∪ B?

Explanation
Answer: B — {1, 2, 3, 4, 5, 6, 7}
UNION (∪) means all elements that appear in A OR B (no duplicates).
Combine both sets and remove repeats: 1,2,3,4,5 from A + 6,7 from B = {1,2,3,4,5,6,7}.
Trick: Many students pick A∩B by accident. Remember: ∪ looks like a bucket collecting everything.
Q 02
SetsEasy

Using the same sets A = {1,2,3,4,5} and B = {3,4,5,6,7},
what is A ∩ B?

Explanation
Answer: C — {3, 4, 5}
INTERSECTION (∩) = elements in BOTH sets simultaneously.
Check each: 3 ∈ A and 3 ∈ B ✓, 4 ✓, 5 ✓. Only {3,4,5}.
Memory: ∩ looks like an upside-down arch — only the overlap passes through.
Q 03
SetsVenn DiagramMedium

In a class of 30 students: 18 study French, 14 study Spanish, and 8 study both.
How many students study neither language?

Formula to use
|A ∪ B| = |A| + |B| − |A ∩ B|
Then: Neither = Total − |A ∪ B|
Explanation
Answer: A — 6
Step 1: |A ∪ B| = 18 + 14 − 8 = 24
Step 2: Neither = 30 − 24 = 6
Common mistake: Forgetting to subtract the overlap (8) leads to 30−32 which is wrong. Always subtract the intersection to avoid double-counting.
Q 04
SetsMedium

The universal set is U = {1,2,3,4,5,6,7,8,9,10} and A = {2,4,6,8}.
What is A' (complement of A)?

Explanation
Answer: B — {1,3,5,7,9,10}
The complement A' = everything in U that is NOT in A.
U = {1,2,3,4,5,6,7,8,9,10}, remove {2,4,6,8} → {1,3,5,7,9,10}.
Don't forget 10! Students often only write odd numbers {1,3,5,7,9} but 10 is even and NOT in A — wait, 10 is not in A, so it IS in A'. Answer B includes 10. ✓
Q 05
SetsTricky

Which of the following is always true for any sets A and B?

Explanation
Answer: B — De Morgan's Law
De Morgan's Laws:
• (A ∪ B)' = A' ∩ B' ← "NOT (A or B)" = "NOT A AND NOT B"
• (A ∩ B)' = A' ∪ B' ← "NOT (A and B)" = "NOT A OR NOT B"
Options A, C, D are only true in special cases (e.g., when A = B or B ⊆ A).
02 — Basic Probability
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Quick Memory Key

P(event) = favourable ÷ total
Always: 0 ≤ P ≤ 1 · P(A) + P(A') = 1
Complementary: P(not A) = 1 − P(A)
Impossible = 0, Certain = 1

Q 06
ProbabilityEasy

A bag contains 5 red, 3 blue, and 2 green marbles.
What is the probability of picking a blue marble at random?

Explanation
Answer: B — 3/10
Total = 5 + 3 + 2 = 10.
P(blue) = 3/10.
Option D (1/3) is wrong because you must use total marbles (10), not just the non-red ones.
Q 07
ProbabilityEasy

The probability it rains tomorrow is 0.35.
What is the probability it does not rain?

Explanation
Answer: B — 0.65
Complementary rule: P(not A) = 1 − P(A) = 1 − 0.35 = 0.65.
Rain and no-rain are complementary events — they cover all possibilities and cannot overlap.
Q 08
ProbabilitySample SpaceMedium

A fair die is rolled once. What is the probability of getting a number greater than 4?

Explanation
Answer: A — 1/3
Sample space = {1,2,3,4,5,6}. "Greater than 4" = {5, 6} → 2 outcomes.
P = 2/6 = 1/3.
Watch out: "greater than 4" does NOT include 4. A common error is writing {4,5,6} giving 3/6 = 1/2 (that's ≥ 4).
03 — Combined Events
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Quick Memory Key

MUTUALLY EXCLUSIVE = can't happen together → P(A ∩ B) = 0
P(A ∪ B) = P(A) + P(B)

INDEPENDENT = one doesn't affect the other
P(A ∩ B) = P(A) × P(B)

GENERAL ADDITION: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Q 09
Combined EventsMedium

Events A and B are mutually exclusive. P(A) = 0.4 and P(B) = 0.25.
Find P(A ∪ B).

Explanation
Answer: B — 0.65
Mutually exclusive → P(A ∩ B) = 0.
P(A ∪ B) = P(A) + P(B) = 0.4 + 0.25 = 0.65.
No need to subtract anything because they can never overlap.
Q 10
Combined EventsTricky

P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.2.
Find P(A ∪ B).

Explanation
Answer: B — 0.70
General addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
= 0.5 + 0.4 − 0.2 = 0.70.
Common mistake: Answer A (0.90) = forgetting to subtract 0.2 (double-counting the overlap).
Q 11
IndependentMedium

A coin is flipped and a die is rolled. These are independent events.
What is P(Heads AND rolling a 6)?

Explanation
Answer: C — 1/12
Independent events: P(A ∩ B) = P(A) × P(B)
P(Heads) = 1/2, P(6) = 1/6
P(Heads AND 6) = 1/2 × 1/6 = 1/12.
Option D is only P(6) — don't forget to multiply by P(Heads) too.
04 — Conditional Probability
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Quick Memory Key

"Given that" = conditional
P(A | B) = P(A ∩ B) / P(B)
Read as: "P of A, given B has happened"
SHRINK the sample space to B, then find A within it.
If independent: P(A|B) = P(A) — knowing B gives no info about A.

Q 12
ConditionalMedium

P(A ∩ B) = 0.12 and P(B) = 0.3.
Find P(A | B).

Explanation
Answer: B — 0.4
P(A|B) = P(A ∩ B) / P(B) = 0.12 / 0.3 = 0.4.
Option A is 0.12 × 0.3 (multiplied instead of divided — common reversal error).
Option D is 0.3 / 0.12 (divided the wrong way).
Q 13
ConditionalTricky

A card is drawn from a standard deck of 52 cards. Given that the card is a heart, what is the probability it is also a face card (J, Q, K)?

Hint
Hearts in a deck: 13 · Face cards that are hearts: J♥, Q♥, K♥ = 3
Explanation
Answer: C — 3/13
Given it's a heart → new sample space = 13 hearts.
Face cards that are hearts = {J♥, Q♥, K♥} = 3.
P(face | heart) = 3/13.
Key insight: The "given" condition shrinks your universe from 52 to 13.
05 — Combinations & Counting
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Quick Memory Key

COMBINATION = choose, order doesn't matter
ⁿCᵣ = n! / (r! × (n−r)!)

PERMUTATION = arrange, order matters
ⁿPᵣ = n! / (n−r)!

Trick: C = "Choose" (no order), P = "Place" (order counts)

Q 14
CombinationsEasy

In how many ways can a committee of 3 students be chosen from a group of 7?

⁷C₃ = 7! / (3! × 4!) = ?
Explanation
Answer: B — 35
⁷C₃ = (7 × 6 × 5) / (3 × 2 × 1) = 210 / 6 = 35.
Option A (21) = ⁷C₂. Option C (210) = ⁷P₃ (permutation, not combination).
Shortcut: (7×6×5)/(3!) — multiply top r terms, divide by r!
Q 15
CombinationsMedium

A pizza shop offers 8 toppings. A customer wants to choose exactly 2 toppings.
How many different pizzas are possible?

Explanation
Answer: C — 28
⁸C₂ = (8 × 7) / (2 × 1) = 56 / 2 = 28.
Option B (56) = ⁸P₂ or 8×7 without dividing — order doesn't matter for pizza toppings! Mushroom+Pepperoni = Pepperoni+Mushroom.
Q 16
CombinationsProbabilityMedium

A box has 4 red and 6 blue balls. Two balls are drawn at random without replacement.
What is the probability that both are red?

Strategy
P(both red) = C(4,2) / C(10,2) ← combinations approach
Explanation
Answer: A — 2/15
C(4,2) = 6 ways to pick 2 reds · C(10,2) = 45 ways to pick any 2
P = 6/45 = 2/15.
OR: P = (4/10) × (3/9) = 12/90 = 2/15 ✓ (sequential method gives same answer).
Without replacement: second draw has only 9 balls total!
06 — Tree Diagrams & Multi-stage
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Quick Memory Key

Multiply along branches (AND = ×)
Add between branches (OR = +)
Each set of branches must sum to 1
With replacement → probabilities stay same
Without replacement → denominator shrinks by 1

Q 17
Tree DiagramMedium

A bag has 3 red and 2 white balls. One ball is drawn, its color noted, then replaced. A second is drawn.
P(one red, one white in any order) = ?

Explanation
Answer: B — 12/25
P(R then W) = 3/5 × 2/5 = 6/25
P(W then R) = 2/5 × 3/5 = 6/25
Total = 6/25 + 6/25 = 12/25
Key: "any order" means TWO paths on the tree — add them!
Q 18
Multi-stageTricky

A student takes two tests. P(pass Test 1) = 0.7. If they pass Test 1, P(pass Test 2) = 0.8. If they fail Test 1, P(pass Test 2) = 0.5.
P(pass both tests) = ?

Explanation
Answer: A — 0.56
P(pass both) = P(pass T1) × P(pass T2 | pass T1) = 0.7 × 0.8 = 0.56.
This is ONE path: Pass T1 → Pass T2. Don't add the other branches; the question asks for "both."
Q 19
Multi-stageMedium

Using the same scenario as Q18, what is the probability the student passes at least one test?

Explanation
Answer: D — 0.91
Use complement: P(at least one pass) = 1 − P(fail both).
P(fail T1) = 0.3 · P(fail T2 | fail T1) = 0.5
P(fail both) = 0.3 × 0.5 = 0.15
P(at least one) = 1 − 0.15 = 0.91
"At least one" = use complement! It's much faster than adding all passing paths.
Q 20
Sets + ProbabilitySynthesis

In a group of 50 students: 30 play football (F), 20 play basketball (B), and 10 play both.
A student is chosen at random. What is the probability the student plays football but NOT basketball?

Explanation
Answer: B — 2/5
"Football only" = F − (F ∩ B) = 30 − 10 = 20 students.
P(F only) = 20/50 = 2/5.
Option A (3/5) confuses P(F) with P(F only). Don't include those who play both!
0/20

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