4.7 Discrete random variables questions for IB Maths AA

35 questions on Discrete random variables, 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	16	4	20
Paper 2	13	1	14
Paper 3HL only	0	1	1
Every paper	29	6	35

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-00423Paper 1 · 15 marks

A spinner is divided into sectors labelled 0, 1, 2 and 3. Let $X$ be the score obtained when the spinner is spun once. The probability distribution of $X$ is given below. Both $p$ and $q$ are constants.

$x$	0	1	2	3
$P (X = x)$	$p$	0.3	$q$	0.2

It is given that $E (X) = 1.4$ .

(a)
Find the value of $p$ and the value of $q$ .
4 marks
(b)
Find $P (X \geq 2 ∣ X \geq 1)$ .
3 marks
(c)
Show that $P (T = 3) = 0.25$ .
3 marks
(d)
Given that $T = 3$ , find the probability that one of the spins scored 0.
2 marks
(e)
Let $F$ be the event "the first spin scores 1" and let $E$ be the event " $T = 3$ ". Determine whether the events $F$ and $E$ are independent. Justify your answer.
3 marks

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Question 2 · AA-00550Paper 2 · 17 marks

A circular dartboard has centre O and radius 25 cm. The bull is the region inside a circle with centre O and radius 5 cm.

The rest of the board is a ring. A sector of the board, with angle 1.2 radians at O, is shaded, and the shaded part of the ring scores 20 points. The bull scores 50 points. The rest of the ring scores 5 points.

Asif throws darts at the board. Every dart hits the board, and the probability that a dart lands in any region of the board is proportional to the area of that region.

(a)
Show that the probability that a dart lands in the bull is 0.04.
2 marks
(b)
Find the area of the part of the ring that scores 20 points.
3 marks
(c)
Find the probability that a dart scores 20 points.
2 marks
(d)
Find the expected score for one dart.
3 marks
(e)(i)
Asif throws 40 darts.
Write down the expected number of darts that land in the bull.
1 mark
(e)(ii)
Find the probability that at least 3 of the 40 darts land in the bull.
3 marks
(f)
The angle of the shaded sector is changed to $θ$ radians. The radii of the board and the bull are not changed.
Find the value of $θ$ for which the expected score for one dart is 12.
3 marks

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Question 3 · AA-00482Paper 3 · HL only · 28 marks

In this question you will compare discrete and continuous models for the score on a spinner.

A fair spinner has $n$ equal sectors, numbered $1, 2, \dots, n$ , where $n \geq 2$ . The score, $X$ , is the number on the sector where the spinner stops, so $P (X = r) = \frac{1}{n}$ for $r = 1, 2, \dots, n$ .

You may use the results $\sum_{r = 1}^{n} r = \frac{n (n + 1)}{2}$ , $\sum_{r = 1}^{n} r^{2} = \frac{n (n + 1) (2 n + 1)}{6}$ and $\sum_{r = 1}^{n} r^{3} = \frac{n^{2} (n + 1)^{2}}{4}$ .

(a)
Consider the case $n = 6$ . Find $E (X)$ and $V a r (X)$ .
3 marks
(b)(i)
Show that, for a spinner with $n$ sectors, $E (X) = \frac{n + 1}{2}$ .
1 mark
(b)(ii)
Show that $V a r (X) = \frac{n^{2} - 1}{12}$ .
4 marks
(c)
In a continuous model, the score is replaced by a random variable $Y$ with probability density function
$g (y) = {\begin{cases} \frac{1}{n}, & \frac{1}{2} \leq y \leq n + \frac{1}{2} \\ 0, & otherwise. \end{cases}$
Show that $E (Y) = \frac{n + 1}{2}$ and $V a r (Y) = \frac{n^{2}}{12}$ .
5 marks
(d)
The value of $X$ can be obtained by rounding $Y$ to the nearest integer. Show that $V a r (Y) - V a r (X)$ does not depend on $n$ , and state its value.
2 marks
(e)(i)
The spinner is replaced by a biased spinner with the same $n$ sectors. Its score $W$ has $P (W = r) = \frac{2 r}{n (n + 1)}$ , for $r = 1, 2, \dots, n$ .
Show that $E (W) = \frac{2 n + 1}{3}$ .
2 marks
(e)(ii)
Show that $V a r (W) = \frac{(n - 1) (n + 2)}{18}$ .
4 marks
(f)
A continuous model for $W$ is the random variable $V$ with probability density function
$h (v) = {\begin{cases} \frac{2 v}{n^{2}}, & 0 \leq v \leq n \\ 0, & otherwise. \end{cases}$
Find $V a r (V)$ in terms of $n$ .
3 marks
(g)
Find $\lim_{n \to \infty} \frac{V a r (W)}{V a r (V)}$ , and explain briefly the significance of this result.
2 marks
(h)
In a game, a player pays $ $c$ to spin the biased spinner with $n = 10$ sectors and then receives $ $W$ .
Find the value of $c$ for which the game is fair.
2 marks

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Question 4 · AA-00457Paper 1 · HL only · 15 marks

A game uses a spinner and three bags, A, B and C.

Bag A contains 2 red balls and 2 white balls.
Bag B contains 3 red balls and 1 white ball.
Bag C contains 4 red balls.

The spinner shows A with probability $\frac{1}{2}$ , B with probability $\frac{1}{3}$ and C with probability $\frac{1}{6}$ . A player spins the spinner once and then takes two balls at random, without replacement, from the bag shown.

Let $X$ be the number of red balls taken.

(a)
Show that $P (X = 0) = \frac{1}{12}$ .
2 marks
(b)
Find $P (X = 1)$ and $P (X = 2)$ .
4 marks
(c)
Find $E (X)$ .
2 marks
(d)
Given that both balls are red, find the probability that they were taken from bag C.
2 marks
(e)
Given that at least one of the balls is red, find the probability that they were taken from bag A.
3 marks
(f)
A player pays $c$ dollars to play the game and receives 3 dollars for each red ball taken. Find the value of $c$ for which the game is fair.
2 marks

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Question 5 · AA-00573Paper 2 · 6 marks

A biased spinner has five sectors numbered 1, 2, 3, 4 and 5. Let $X$ be the number obtained when the spinner is spun once. The probability distribution of $X$ is given in the following table, where $k$ is a constant.

$x$	1	2	3	4	5
$P (X = x)$	$0.12$	$2 k$	$0.31$	$k$	$0.09$

(a)
Find the value of $k$ .
2 marks
(b)
Find $P (X > 3)$ .
1 mark
(c)
Find $E (X)$ .
2 marks
(d)
The spinner is spun 60 times. Find the expected number of times the spinner lands on 4.
1 mark

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Question 6 · AA-00462Paper 1 · HL only · 8 marks

The table below shows the probability distribution of a discrete random variable $X$ . The values $p$ and $q$ are unknown.

$x$	$0$	$1$	$2$	$4$
$P (X = x)$	$0.2$	$p$	$q$	$0.1$

It is given that $E (X) = 1.5$ .

(a)
Find the value of $p$ and the value of $q$ .
3 marks
(b)
Find $V a r (X)$ .
3 marks
(c)
Find $E (4 - 3 X)$ and $V a r (4 - 3 X)$ .
2 marks

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