$P(C)=0.6$, $P(B)=0.45$, $P(C \cup B)=0.85$. 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 $0.444\ldots$).

The medical records of a group of 30 children show that 24 had measles ($M$) and 12 had mumps ($P$). 4 children had neither. Let $x$ be the number of children who had both illnesses. Set up a linear equation to find the value of $x$. Given that a randomly selected child had measles, what is the exact probability they also had mumps?

The number of children who had at least one illness is $30 - 4 = 26$. Using $n(M \cup P) = n(M) + n(P) - n(M \cap P)$: $26 = 24 + 12 - x$ M1 $26 = 36 - x \implies x = 10$ A1 Restrict to the 24 children who had measles: $P(P \mid M) = \frac{n(P \cap M)}{n(M)} = \frac{10}{24}$ M1 $= \frac{5}{12}$ A1

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IB Maths AI SL · Unit 4: Statistics and Probability

IB Maths AI SL Venn Diagrams & Hard Probability Questions

Updated October 2026

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, showing the working that earns the marks — 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 →

What you need to know

P(A|B) = P(A ∩ B)/P(B) — but SL AI expects you to build the tree diagram and read off the answer, not manipulate the formula abstractly. Conditional probability with tree diagrams overview →

Perhaps SL AI's single most examined topic. Know the difference between GoF and independence tests, calculate expected values, and interpret the p-value against the significance level. Chi-squared test — goodness of fit and independence overview →

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: $P (A) = \frac{n (A)}{n (U)}$
Conditional probability: $P (A ∣ B) = \frac{P (A \cap B)}{P (B)}$
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 →

Explained, with a worked example: Probability 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?

Solution

5 steps · 1 to fill in

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 $P (C ∣ B)$ exactly.

Solution

5 steps

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 ( $3 R, 2 B$ ); tails $\to$ Box B ( $4 R, 1 B$ ). Given a red marble is drawn, find $P (Box A ∣ Red)$ .

Solution

5 steps · 1 to fill in

Examiner tip: Reverse conditional probability = specific successful path

\div

sum of all successful paths.

Try these IB Maths AI SL Venn diagrams & hard probability questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · medium · 4 marks · Paper 1

The medical records of a group of 30 children show that 24 had measles ( $M$ ) and 12 had mumps ( $P$ ). 4 children had neither.

Let $x$ be the number of children who had both illnesses.

Venn diagram: universal set U = 30 with overlapping sets M (measles) and P (mumps), x children in both, 4 in neither

Set up a linear equation to find the value of $x$ .
Given that a randomly selected child had measles, what is the exact probability they also had mumps?

Full worked solution Free, no sign-in

The number of children who had at least one illness is $30 - 4 = 26$ .
Using $n (M \cup P) = n (M) + n (P) - n (M \cap P)$ :
$26 = 24 + 12 - x$ M1
$26 = 36 - x ⟹ x = 10$ A1
Restrict to the 24 children who had measles:
$P (P ∣ M) = \frac{n (P \cap M)}{n (M)} = \frac{10}{24}$ M1
$= \frac{5}{12}$ A1

Every question here has a full worked solution. See them all →

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All 15 Venn diagrams & hard probability questions with mark schemes →

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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.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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IB Maths AI SL Venn Diagrams & Hard Probability Questions

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What you need to know

What's examined in AI SL Venn diagrams & hard probability

Key formulas

Venn Diagrams & Hard Probability worked examples

Worked example 1: Mutually exclusive vs independent · easy

Solution

Worked example 2: Conditional probability from a Venn diagram · medium

Solution

Worked example 3: Bayes' theorem with a biased coin · hard

Solution

Try these IB Maths AI SL Venn diagrams & hard probability questions

Question 1 · medium · 4 marks · Paper 1

Full worked solution Free, no sign-in

Revise Venn diagrams & hard probability for the exam

FAQ

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