📐 IB Mathematics · Grade 9

Probability &
Combinatorics

20 essential questions covering sets, counting, and probability — designed to sharpen your instincts before exams.

20
Questions
4
Topics
★★☆
Difficulty
0 of 20 answered 0%
🔵
Set Theory
Union · Intersection · Complement · Venn Diagrams
🧠
Quick Memory Point
A ∪ B — "U for Union = glUe everything together"
∪ means ALL elements in A or B (or both). ∩ means ONLY what's in the middle. A' (complement) means everything NOT in A.
UNION = ∪ = U-shape = "U get ALL" INTERSECTION = ∩ = ∩-shape = "n-shape = iN the middle"

Key Formulas — Sets

Inclusion-Exclusion
\(|A \cup B| = |A| + |B| - |A \cap B|\)
Complement
\(|A'| = |U| - |A|\)
De Morgan's Law
\((A \cup B)' = A' \cap B'\)
Subset
\(A \subseteq B \Rightarrow A \cap B = A\)
1
Sets · Basic ● Easy
In a class of 30 students, 18 play soccer, 12 play basketball, and 5 play both. How many students play at least one sport?
Think: Don't add 18 + 12 = 30 directly! The 5 who play both are counted TWICE. Use:
\(|A \cup B| = |A| + |B| - |A \cap B|\)
A25
B30
C20
D35
2
Sets · Complement ● Easy
Universal set \(U = \{1,2,3,4,5,6,7,8\}\), \(A = \{2,4,6,8\}\). Which is \(A'\)?
A\(\{2,4,6,8\}\)
B\(\{1,3,5,7\}\)
C\(\{1,2,3,4,5,6,7,8\}\)
D\(\emptyset\)
3
Sets · De Morgan ● Medium
Which of the following is equivalent to \((A \cap B)'\)?
Trap: Students often confuse \((A \cap B)'\) with \(A' \cap B'\). Remember De Morgan!
A\(A' \cap B'\)
B\(A' \cup B'\)
C\(A \cup B\)
D\(A \cap B\)
4
Sets · Venn ● Medium
In a group of 50 people: 30 like coffee, 25 like tea, and 10 like neither. How many like both coffee and tea?
Step 1: Those who like at least one = 50 − 10 = 40
Step 2: Apply inclusion-exclusion: \(|A \cup B| = |A| + |B| - |A \cap B|\)
A5
B15
C10
D20
5
Sets · Subset ● Easy
How many subsets does the set \(\{a, b, c\}\) have?
Memory: A set with \(n\) elements has \(2^n\) subsets (include the empty set ∅ and the set itself!)
A6
B8
C3
D9
🟣
Counting Principles
Multiplication Rule · Permutations · Arrangements
🔢
Quick Memory Point
Multiplication Rule: "AND = Multiply, OR = Add"
Choose shirt AND pants → multiply choices.
Take bus OR walk → add the options.
AND = × OR = + Factorial n! = n×(n−1)×...×1

Key Formulas — Counting

Permutation (order matters)
\(P(n,r) = \dfrac{n!}{(n-r)!}\)
Combination (order doesn't matter)
\(C(n,r) = \dbinom{n}{r} = \dfrac{n!}{r!(n-r)!}\)
Factorial
\(n! = n \times (n{-}1) \times \cdots \times 1\)
Multiply Rule
Task A (m ways) AND B (n ways) \(= m \times n\)
6
Counting · Multiplication ● Easy
A restaurant has 4 starters, 6 mains, and 3 desserts. How many different 3-course meals are possible?
A13
B72
C36
D48
7
Counting · Permutation ● Medium
In how many ways can 5 people be seated in a row of 5 chairs?
Think: 5 choices for seat 1, 4 for seat 2, 3 for seat 3 ... = \(5!\)
A25
B120
C60
D20
8
Counting · P(n,r) ● Medium
How many 3-letter codes can be made from the letters A, B, C, D, E if no letter is repeated? (Order matters)
Formula: \(P(5,3) = \dfrac{5!}{(5-3)!} = \dfrac{5!}{2!} = 5 \times 4 \times 3\)
A10
B120
C60
D30
9
Counting · Combination ● Medium
A committee of 3 students is chosen from a group of 7. How many different committees are possible? (Order does NOT matter)
Key trap: Committee {Amy, Bob, Carl} = {Carl, Amy, Bob}. Order doesn't matter → use Combination:
\(\binom{7}{3} = \dfrac{7!}{3! \cdot 4!}\)
A210
B35
C21
D42
10
Counting · Tricky! ● Hard
From 4 boys and 3 girls, a team of 3 is selected. How many ways can you pick at least one girl?
Smart trick: "At least one" = Total − None
Total teams \(= \binom{7}{3}\). Teams with NO girls \(= \binom{4}{3}\)
A35
B31
C21
D28
🟡
Basic Probability
Classical · Complementary · Combined Events
🎲
Quick Memory Point
P(A) = Favourable ÷ Total
Probability is always between 0 and 1. P = 0 → impossible. P = 1 → certain.
P(not A) = 1 − P(A) ← this saves you so much time!
P(A') = 1 − P(A) 0 ≤ P ≤ 1 P(certain) = 1

Key Formulas — Probability

Classical Probability
\(P(A) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}\)
Complement Rule
\(P(A') = 1 - P(A)\)
Addition Rule
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
Mutually Exclusive
\(P(A \cup B) = P(A) + P(B)\)
11
Probability · Basic ● Easy
A bag has 3 red, 5 blue, and 2 green marbles. One marble is picked at random. What is the probability it is blue?
A\(\dfrac{1}{2}\)
B\(\dfrac{5}{10}\)
C\(\dfrac{3}{10}\)
D\(\dfrac{2}{5}\)
12
Probability · Complement ● Easy
The probability of rain tomorrow is 0.35. What is the probability it does not rain?
A0.35
B0.65
C0.70
D1.35
13
Probability · Dice ● Easy
A fair die is rolled. What is the probability of getting a number greater than 4?
A\(\dfrac{1}{3}\)
B\(\dfrac{1}{2}\)
C\(\dfrac{2}{3}\)
D\(\dfrac{1}{6}\)
14
Probability · Addition Rule ● Medium
P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2. Find P(A ∪ B).
Trap: Don't forget to subtract the overlap!
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
A0.9
B0.7
C0.2
D1.1
15
Probability · Cards ● Medium
A card is drawn from a standard deck of 52. What is the probability it is a King or a Heart?
Note: The King of Hearts is in BOTH groups — count it only once!
Hearts = 13, Kings = 4, King of Hearts = 1 (overlap)
A\(\dfrac{17}{52}\)
B\(\dfrac{16}{52}\)
C\(\dfrac{4}{13}\)
D\(\dfrac{1}{4}\)
🟢
Conditional & Independent Probability
P(A|B) · Independent Events · Tree Diagrams
🌲
Quick Memory Point
Conditional: P(A|B) = "Probability of A, given B already happened"
Think of it as: shrink your sample space to only B outcomes, then find A within it.
P(A|B) = P(A∩B) ÷ P(B) Independent: P(A∩B) = P(A)×P(B) Dependent: P(B|A) ≠ P(B)

Key Formulas — Conditional & Independence

Conditional Probability
\(P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\)
Independent Events
\(P(A \cap B) = P(A) \cdot P(B)\)
Test for Independence
\(P(A \mid B) = P(A)\)
Dependent (Multiplication)
\(P(A \cap B) = P(A) \cdot P(B \mid A)\)
16
Conditional · Basic ● Medium
P(A) = 0.6, P(B) = 0.5, P(A ∩ B) = 0.3. Find P(A | B).
A0.5
B0.6
C0.3
D0.18
17
Independence · Test ● Medium
P(A) = 0.4, P(B) = 0.3, P(A ∩ B) = 0.12. Are A and B independent?
Check: If independent, then \(P(A) \times P(B)\) must equal \(P(A \cap B)\).
\(0.4 \times 0.3 = ?\) Compare with 0.12.
AYes — because P(A∩B) = P(A)·P(B)
BNo — because P(A∩B) ≠ P(A)·P(B)
CYes — because P(A|B) = P(B)
DCannot be determined
18
Probability · Two Events ● Medium
A bag has 4 red and 6 blue balls. Two balls are drawn without replacement. What is the probability both are red?
Without replacement means the second draw is affected by the first!
\(P(\text{both red}) = P(\text{1st red}) \times P(\text{2nd red} \mid \text{1st red})\)
A\(\dfrac{16}{100}\)
B\(\dfrac{12}{90}\)
C\(\dfrac{4}{15}\)
D\(\dfrac{2}{15}\)
19
Probability · Tree Diagram ● Hard
A biased coin has P(Head) = 0.6. It is flipped twice. What is the probability of getting exactly one Head?
Two ways: HT or TH
P(HT) = 0.6 × 0.4, P(TH) = 0.4 × 0.6
Add them: both are mutually exclusive outcomes.
A0.36
B0.48
C0.24
D0.50
20
Probability · Hardest! ● Hard
A class has 15 students: 8 study French, 6 study Spanish, and 3 study both. A student is picked at random. Given they study French, what is the probability they also study Spanish?
Conditional!: We already know the student is in French → shrink sample space to 8.
Of those 8, how many also do Spanish?
A\(\dfrac{3}{15}\)
B\(\dfrac{3}{8}\)
C\(\dfrac{3}{6}\)
D\(\dfrac{6}{15}\)
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