IB Mathematics · Grade 9

Probability & Sets

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20 Questions · Self-Study Edition

Master Probability
& Set Theory

Carefully crafted questions on the topics students get wrong the most. Choose your answer — instant feedback included.

Sets & Notation Venn Diagrams Basic Probability Combined Events Combinations Conditional Prob.
Part 1 · Sets & Venn Diagrams
01
Set Notation Easy
Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}.
What is A ∩ B?
⚡ MEMORY POINT
∩ = INTERSECTION = "AND" = OVERLAP
A ∩ B → elements in BOTH A and B
02
Union of Sets Easy
Let A = {a, b, c, d} and B = {c, d, e, f}.
How many elements does A ∪ B contain?
⚡ MEMORY POINT
∪ = UNION = "OR" = EVERYTHING
|A ∪ B| = |A| + |B| − |A ∩ B|
03
Complement & Venn Diagram Medium
In a class of 30 students, 18 study French (F) and 14 study Spanish (S). 6 students study both. How many students study neither French nor Spanish?
⚡ MEMORY POINT
NEITHER = Total − |F ∪ S|
|F ∪ S| = |F| + |S| − |F ∩ S| = 18 + 14 − 6
04
Complement Notation Easy
Universal set U = {1, 2, 3, 4, 5, 6, 7, 8} and A = {2, 4, 6, 8}.
What is A' (the complement of A)?
⚡ MEMORY POINT
A' = "NOT A" = everything in U but NOT in A
A' = U \ A → odd numbers here!
05
Subsets Medium
How many subsets does the set A = {p, q, r} have?
(Include the empty set ∅ and A itself)
⚡ MEMORY POINT
Subsets formula: 2ⁿ where n = number of elements
|A| = 3 → 2³ = 8 subsets total
Part 2 · Basic Probability
06
Sample Space Easy
A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?
⚡ MEMORY POINT
P(event) = favorable outcomes ÷ total outcomes
Numbers > 4: {5, 6} → 2 out of 6
07
Complementary Probability Easy
The probability that it rains tomorrow is 0.35.
What is the probability that it does NOT rain?
⚡ MEMORY POINT
P(NOT A) = 1 − P(A)
Always: P(A) + P(A') = 1
08
Mutually Exclusive Events Medium
Events A and B are mutually exclusive. P(A) = 0.3 and P(B) = 0.4.
Find P(A ∪ B).
⚡ MEMORY POINT
MUTUALLY EXCLUSIVE → cannot happen together → P(A ∩ B) = 0
P(A ∪ B) = P(A) + P(B) when mutually exclusive
09
Addition Rule — Tricky! Medium
P(A) = 0.5, P(B) = 0.4, and P(A ∩ B) = 0.2.
What is P(A ∪ B)?
⚡ MEMORY POINT
GENERAL ADDITION RULE (non-mutually exclusive):
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
10
Independent Events — Classic Trap Medium
A coin is flipped and a die is rolled. What is the probability of getting Heads AND rolling a 3?
⚡ MEMORY POINT
INDEPENDENT EVENTS → multiply probabilities
P(A AND B) = P(A) × P(B)
Part 3 · Conditional Probability
11
Conditional Probability Medium
P(A) = 0.6, P(B) = 0.5, P(A ∩ B) = 0.3.
Find P(A | B) — the probability of A given B has occurred.
⚡ MEMORY POINT
CONDITIONAL: "given" = divide by the condition
P(A|B) = P(A ∩ B) / P(B)
12
Tree Diagram / Dependent Events Medium
A bag contains 5 red and 3 blue balls. One ball is drawn and NOT replaced. A second ball is then drawn. What is the probability that both balls are red?
⚡ MEMORY POINT
WITHOUT replacement → DEPENDENT → denominator decreases!
P(R₁ ∩ R₂) = (5/8) × (4/7)
13
Testing Independence — Very Tricky! Hard
P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2.
Are events A and B independent?
⚡ MEMORY POINT
TEST for independence: check if P(A∩B) = P(A) × P(B)
0.4 × 0.5 = 0.20 → equals P(A∩B)? → YES → Independent!
Part 4 · Combinations & Counting
14
Combinations — nCr Easy
How many ways can you choose 2 students from a group of 5?
(Order does NOT matter)
⚡ MEMORY POINT
COMBINATION = "choose" = ORDER DOESN'T MATTER
C(n,r) = n! / [r! × (n−r)!] → C(5,2) = 5!/(2!×3!)
15
Permutation vs Combination — Classic Confusion Medium
In how many ways can a President, Vice President, and Secretary be elected from 8 candidates?
(Each person gets a different role — order matters!)
⚡ MEMORY POINT
PERMUTATION = ORDER MATTERS → use P(n,r)
P(n,r) = n! / (n−r)! → P(8,3) = 8 × 7 × 6
16
Combinations in Probability Medium
A committee of 3 is chosen from 4 men and 3 women. What is the probability that the committee contains exactly 2 women?
⚡ MEMORY POINT
P = favorable combos ÷ total combos
Favorable: C(3,2)×C(4,1) | Total: C(7,3)
Part 5 · Challenge Problems
17
Venn — Three Sets Hard
In a survey of 50 students: 30 like Math (M), 25 like Science (S), 20 like English (E). 10 like M∩S, 8 like M∩E, 7 like S∩E, and 4 like all three. How many students like only Math?
⚡ MEMORY POINT
3-Set Venn: fill FROM THE CENTER outward!
Only M = |M| − (M∩S only) − (M∩E only) − (all three)
18
Conditional from Venn — Sneaky! Hard
From a group of 40 people: 24 own a dog (D), 18 own a cat (C), 8 own both. One person is chosen at random. Given they own a cat, what is the probability they also own a dog?
⚡ MEMORY POINT
"Given cat" = your new sample space is only cat owners (18)
P(D|C) = P(D ∩ C) / P(C) = 8/18
19
C(n,r) Identity — Mind Trick Hard
Which of the following is always true?
⚡ MEMORY POINT
Symmetry of combinations: C(n,r) = C(n, n−r)
e.g. C(10,3) = C(10,7) → choosing 3 ≡ leaving out 7
20
Combined Probability — Final Boss 🔥 Hard
A box has 6 red and 4 blue balls. Two balls are drawn without replacement. What is the probability of getting one red AND one blue (in any order)?
⚡ MEMORY POINT
"Any order" = add both arrangements: RB + BR
P(RB) + P(BR) = (6/10)(4/9) + (4/10)(6/9)
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