IB Maths AI SL · Unit 4: Statistics and Probability
IB Maths AI SL Venn Diagrams & Hard Probability Questions
Exam-style IB Maths AI SL venn diagrams & hard probability questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 15 questions
- Paper 1: 6
- Paper 2: 9
- 2 medium
- 10 hard
- 3 very hard
- 3 worked examples
Practise Venn Diagrams & Hard Probability questions →
AI SL formula booklet
What's examined in AI SL venn diagrams & hard probability
The question bank covers these venn diagrams & hard probability question types (number of questions in brackets):
- Venn Diagram Construction (11)
- Venn Diagram Equations (2)
- Probability Laws & Set Notation (2)
Key formulas
- Probability of an event
- \(P(A) = \dfrac{n(A)}{n(U)}\)
- Conditional probability
- \(P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\)
- Independent events
- \(P(A \cap B) = P(A)\,P(B)\)
- Complementary events
- \(P(A') = 1 - P(A)\)
- Combined events
- \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
- Mutually exclusive events
- \(P(A \cap B) = 0\)
In the same notation as the IB formula booklet. All AI SL formulas →
Venn Diagrams & Hard Probability worked examples
Worked example 1: Mutually exclusive vs independent · easy
$P(A)=0.5$, $P(B)=0.4$, $P(A \cap B)=0.2$. Are $A$ and $B$ mutually exclusive? Independent?
1. Mutually exclusive: requires $P(A \cap B)=0$.
2. Observe: $0.2 \ne 0$, so $A$ and $B$ are not mutually exclusive.
3. Independence test: $P(A \cap B) = P(A) \times P(B)$?
4. Product: $0.5 \times 0.4 = 0.2$.
5. Compare: $0.2 = 0.2 \Rightarrow A$ and $B$ are independent.
Examiner tip: Never assume independence just because events aren't mutually exclusive (or vice versa). Test $P(A\cap B)=P(A)P(B)$ every time.
Worked example 2: Conditional probability from a Venn diagram · medium
$P(C)=0.6$, $P(B)=0.45$, $P(C \cup B)=0.85$. Find $P(C \mid B)$ exactly.
1. Formula: $P(C \mid B) = \frac{P(C \cap B)}{P(B)}$.
2. Addition rule: $P(C \cup B) = P(C) + P(B) - P(C \cap B)$.
3. Substitute: $0.85 = 0.6 + 0.45 - P(C \cap B)$.
4. Solve: $P(C \cap B) = 0.20$.
5. Divide: $\frac{0.20}{0.45} = \mathbf{\tfrac{4}{9}}$ (or $0.444\ldots$).
Examiner tip: "Given that" shrinks the denominator from $1.0$ down to the probability of the condition (here $0.45$).
Worked example 3: Bayes' theorem with a biased coin · hard
$P(H)=0.6$; heads $\to$ Box A ($3R, 2B$); tails $\to$ Box B ($4R, 1B$). Given a red marble is drawn, find $P(\text{Box A} \mid \text{Red})$.
1. Formula: $P(\text{A} \mid \text{R}) = \frac{P(\text{A} \cap \text{R})}{P(\text{R})}$.
2. Box A path: $0.6 \times \tfrac{3}{5} = 0.36$.
3. Box B path: $0.4 \times \tfrac{4}{5} = 0.32$.
4. Total $P(\text{R})$: $0.36 + 0.32 = 0.68$.
5. Divide: $\frac{0.36}{0.68} = \mathbf{\tfrac{9}{17}}$ (or $0.529$).
Examiner tip: Reverse conditional probability = specific successful path $\div$ sum of all successful paths.
FAQ
How many IB Maths AI SL venn diagrams & hard probability questions are there?
There are 15 exam-style venn diagrams & hard probability questions in the AI SL question bank (Paper 1: 6 · Paper 2: 9), graded 10 hard, 3 very hard, 2 medium. Every question has a full IB-style mark scheme (M, A and R marks).
Is venn diagrams & hard probability on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 6 · Paper 2: 9. Practise with your GDC — AI papers expect calculator methods throughout.
Where can I get the mark schemes?
Open the AI SL Unit 4 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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