4.6 Venn diagrams, tree diagrams and combined events questions for IB Maths AA

78 questions on Venn diagrams, tree diagrams and combined events, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
Part of 4 Statistics and probability
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	40	8	48
Paper 2	28	1	29
Paper 3HL only	0	1	1
Every paper	68	10	78

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AA-00409Paper 1 · 6 marks

Consider two events $A$ and $B$ such that $P (A^{'}) = \frac{3}{5}$ , $P (B ∣ A) = \frac{1}{4}$ and $P (A \cup B) = \frac{3}{4}$ .

(a)
Find $P (A \cap B)$ .
2 marks
(b)
Find $P (B)$ .
2 marks
(c)
Find $P (B ∣ A^{'})$ .
2 marks

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Question 2 · AA-00411Paper 2 · 7 marks

On one day, 250 visitors to a museum were asked whether they hired an audio guide ( $G$ ). Each visitor holds exactly one type of ticket: adult, student ( $S$ ) or senior. The results are shown in the following table.

	Audio guide ( $G$ )	No audio guide ( $G^{'}$ )	Total
Adult	48	72	120
Student ( $S$ )	$x$	$y$	75
Senior	22	33	55

A visitor is selected at random from these 250 visitors.

(a)
Given that $P (G ∣ S) = 0.36$ , find the value of $x$ and the value of $y$ .
2 marks
(b)
Find the probability that the visitor holds a student ticket, given that the visitor hired an audio guide.
2 marks
(c)
Determine whether the events $G$ and $S$ are independent. Justify your answer.
3 marks

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Question 3 · AA-00459Paper 3 · HL only · 25 marks

This question asks you to investigate whether a contestant in a game should change their choice after receiving information from a host.

In a game, a prize is placed behind one of three closed doors, numbered 1, 2 and 3. The prize is equally likely to be behind each door. A contestant chooses door 1. The host, who knows where the prize is, then opens one of the other two doors, always choosing a door that does not hide the prize. The contestant may then either stay with door 1 or switch to the other closed door.

If the prize is behind door 1, the host opens door 3 with probability $q$ and door 2 with probability $1 - q$ , where $0 \leq q \leq 1$ .

Let $D_{i}$ be the event that the prize is behind door $i$ , for $i = 1, 2, 3$ , and let $H_{3}$ be the event that the host opens door 3.

(a)
Write down $P (H_{3} ∣ D_{1})$ , $P (H_{3} ∣ D_{2})$ and $P (H_{3} ∣ D_{3})$ .
2 marks
(b)
Show that $P (H_{3}) = \frac{1 + q}{3}$ .
2 marks
(c)(i)
Show that $P (D_{1} ∣ H_{3}) = \frac{q}{1 + q}$ .
2 marks
(c)(ii)
Find $P (D_{2} ∣ H_{3})$ in terms of $q$ .
2 marks
(d)
Consider the case $q = \frac{1}{2}$ . The host opens door 3. Find the probability that the contestant wins the prize if they switch to door 2.
2 marks
(e)
Show that, when the host opens door 3, switching to door 2 is never less likely to win the prize than staying with door 1. State the value of $q$ for which the two choices are equally likely to win.
3 marks
(f)
The contestant decides, before the game starts, always to switch. By finding $P (D_{3} ∣ H_{2})$ , where $H_{2}$ is the event that the host opens door 2, show that the probability that the contestant wins the prize is $\frac{2}{3}$ for every value of $q$ .
4 marks
(g)
Find the probability that the game is abandoned.
2 marks
(h)
This host opens door 3 and it does not hide the prize. Show that the probability that the prize is behind door 1 is now $\frac{1}{2}$ .
4 marks
(i)
By comparing your answers to parts (d) and (h), explain briefly why the host's knowledge matters to the contestant.
2 marks

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Question 4 · AA-00410Paper 1 · 5 marks

In a class of 40 students, 10 students belong to the chess club ( $C$ ) and 16 students play in the school orchestra ( $O$ ). Four students belong to the chess club and also play in the orchestra.

A student is selected at random from the class.

(a)
Determine whether the events $C$ and $O$ are independent. Justify your answer.
3 marks
(b)
Find the probability that the student belongs to the chess club, given that the student does not play in the orchestra.
2 marks

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Question 5 · AA-00424Paper 2 · 16 marks

A council surveyed 400 households in a town. Each household was asked whether it has a heat pump ( $H$ ) and whether it has solar panels ( $S$ ). The results are shown in the following table.

	Solar panels ( $S$ )	No solar panels ( $S^{'}$ )	Total
Heat pump ( $H$ )	36	$x$	$36 + x$
No heat pump ( $H^{'}$ )	84	$280 - x$	$364 - x$
Total	120	280	400

A household is selected at random from the 400 households.

(a)
Write down $P (S)$ .
1 mark
(b)
Given that $P (H ∣ S^{'}) = 0.15$ , find the value of $x$ .
2 marks
(c)
Find the probability that the household has solar panels, given that it has a heat pump.
2 marks
(d)
Determine whether the events $H$ and $S$ are independent. Justify your answer.
3 marks
(e)
In a survey in a neighbouring town, 36 households have a heat pump and solar panels, 84 have solar panels but no heat pump, and 238 have neither. In this town, $H$ and $S$ are independent. Find the number of households in this survey that have a heat pump but no solar panels.
3 marks
(f)
Find the probability that exactly one of the three households has a heat pump.
3 marks
(g)
Twenty-five households in the town are chosen at random. Find the probability that at least six of them have a heat pump.
2 marks

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Question 6 · AA-00412Paper 1 · 7 marks

Consider two independent events $A$ and $B$ such that $P (B) = 2 P (A)$ and $P (A \cup B) = \frac{5}{8}$ .

Let $P (A) = p$ .

(a)
Show that $16 p^{2} - 24 p + 5 = 0$ .
3 marks
(b)
Find the value of $p$ , giving a reason for your answer.
2 marks
(c)
Find $P (A^{'} ∣ B)$ .
2 marks

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