The probability distribution of a discrete random variable $X$ is given by the formula: \[P(X = x) = \frac{x}{k} \quad \text{for } x \in \{1, 2, 3, 4\}\] where $k$ is a positive constant. Find the exact value of $k$ . Find the expected value, $E(X)$ .

Sum of probabilities $= 1$ . $\frac{1}{k} + \frac{2}{k} + \frac{3}{k} + \frac{4}{k} = 1$ . (M1) $\frac{10}{k} = 1 \implies k = 10$ . A1 $E(X) = \sum x P(X=x) = 1\left(\frac{1}{10}\right) + 2\left(\frac{2}{10}\right) + 3\left(\frac{3}{10}\right) + 4\left(\frac{4}{10}\right)$ . (M1) $E(X) = \frac{1 + 4 + 9 + 16}{10}$ . (A1) $E(X) = \frac{30}{10} = 3$ . A1

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IB Maths AA SL · Unit 4: Statistics and Probability

IB Maths AA SL Discrete Random Variables Questions

Q: How many IB Maths AA SL discrete random variables questions are there?

There are 50 exam-style discrete random variables questions in the AA SL question bank (Paper 1: 32 · Paper 2: 18), graded 14 easy, 20 medium, 11 hard, 5 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is discrete random variables on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 32 · Paper 2: 18. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Q: Where can I get the mark schemes?

Open the AA 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.

Updated October 2026

Exam-style IB Maths AA SL discrete random variables 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.

50 questions
Paper 1: 32
Paper 2: 18
14 easy
20 medium
11 hard
5 very hard
3 worked examples

Practise Discrete Random Variables questions →

🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

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1 of 5

Question 1 of 5
Question 2 of 5
Question 3 of 5
Question 4 of 5
Question 5 of 5

Start with one question on discrete random variables. Stuck? Take a hint. No sign-in needed.

Find

E (X)

for the distribution in the table.

$x$	-1	0	3
$P (X = x)$	$\frac{2}{5}$	$\frac{2}{5}$	$\frac{1}{5}$

Choose your answer · keys 1–4

\frac{2}{3}

1

2.2

0.2

Find

E (X)

for the distribution in the table.

$x$	1	3	5	7
$P (X = x)$	$\frac{1}{8}$	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{1}{8}$

Choose your answer · keys 1–4

19

1

4

16

Find

E (X)

for the distribution in the table.

$x$	10	20	50
$P (X = x)$	$\frac{7}{10}$	$\frac{1}{5}$	$\frac{1}{10}$

Choose your answer · keys 1–4

400

\frac{80}{3}

1

16

Find

E (X)

for the distribution in the table.

$x$	1	2	3	4
$P (X = x)$	$\frac{1}{10}$	$\frac{1}{5}$	$\frac{3}{10}$	$\frac{2}{5}$

Choose your answer · keys 1–4

1

10

3

2.5

$x$	0	1	2
$P (X = x)$	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{4}$

What you need to know

n trials, probability p of success. SL AA expects you to know both the mean (np) and variance (np(1-p)) plus how to use the GDC's binomialCDF function. Binomial distribution — B(n,p) overview →

What's examined in AA SL discrete random variables

The question bank covers these discrete random variables question types (number of questions in brackets):

Discrete Prob. Distributions (32)
Binomial Probabilities (12)
Binomial Parameters (6)

Key formulas

Expected value (discrete): $E (X) = \sum x P (X = x)$
Mean & variance of a binomial: $E (X) = n p, Var (X) = n p (1 - p)$
Standardising a normal variable: $Z = \frac{X - μ}{σ}$

In the same notation as the IB formula booklet. All AA SL formulas →

Explained, with a worked example: Binomial distribution

Where students lose marks on discrete random variables

Examiner's noteBinomial and normal: "at least", inequalities, rounding bounds and which distribution

Where marks are lost: Recurring errors: P(X ≥ 1) written as 1 − P(X ≤ 1); forgetting to reverse an inequality after dividing by a negative logarithm; an off-by-one slip when turning a strict inequality for a whole-number variable into a cumulative probability; a normal probability found for a single value instead of between the rounding bounds; the binomial used where the normal was needed (or the reverse); an inverse-normal value taken from the wrong tail; and μ and σ from two probabilities solved by hand.

Examiners flagged this in 4 exam series (May 2024, November 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Discrete Random Variables worked examples

Worked example 1: Finding probability constants and expected value · easy

A biased 4-sided die has $P (X = 1) = P (X = 2) = P (X = 3) = p$ and $P (X = 4) = \frac{1}{2}$ . Find $p$ and $E (X)$ .

Solution

3 steps

Examiner tip: Multiply each outcome by its probability BEFORE adding. Averaging the outcomes alone is a common mistake.

Worked example 2: Solving for a missing prize in a fair game · medium

A game costs $€ 3$ . Prize distribution: $P (W = 0) = 0.4, P (W = 2) = 0.3, P (W = 5) = 0.2, P (W = A) = 0.1$ . Given the game is fair, find $A$ .

Solution

4 steps

Examiner tip: A "fair game" makes expected NET gain zero, which is equivalent to

E (payout) = cost

Worked example 3: Conditional binomial probability · hard

Defect rate $12 %$ , sample $n = 50$ . Given the inspector finds fewer than 8 defective chips, find $P (D = 5 ∣ D < 8)$ .

Solution

5 steps

Examiner tip: "Fewer than 8" for a discrete variable is

\leq 7

, NOT

\leq 8

Try these IB Maths AA SL discrete random variables 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 · easy · 4 marks · Paper 2

A fair coin is tossed 20 times and the number of times it lands heads up is recorded as the random variable $X$ .

Find the expected number of times that the coin will land heads up.
Use your graphic display calculator to find the probability that the coin lands heads up exactly 15 times.

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

The probability distribution of a discrete random variable $X$ is given by the formula: $P (X = x) = \frac{x}{k} for x \in {1, 2, 3, 4}$ where $k$ is a positive constant.

Find the exact value of $k$ .
Find the expected value, $E (X)$ .

Full worked solution Free, no sign-in

Sum of probabilities $= 1$ .
$\frac{1}{k} + \frac{2}{k} + \frac{3}{k} + \frac{4}{k} = 1$ . (M1)
$\frac{10}{k} = 1 ⟹ k = 10$ . A1
$E (X) = \sum x P (X = x) = 1 (\frac{1}{10}) + 2 (\frac{2}{10}) + 3 (\frac{3}{10}) + 4 (\frac{4}{10})$ . (M1)
$E (X) = \frac{1 + 4 + 9 + 16}{10}$ . (A1)
$E (X) = \frac{30}{10} = 3$ . A1

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

Question 3 · hard · 5 marks · Paper 2

A company manufactures computer chips. It is known that $12 %$ of the chips produced are defective. A quality control inspector takes a random sample of 50 chips. Let the random variable $D$ represent the number of defective chips in the sample.

Find the probability that the inspector finds fewer than 8 defective chips.
Given that the inspector finds fewer than 8 defective chips, find the conditional probability that they found exactly 5 defective chips.

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AA SL discrete random variables questions are there?

There are 50 exam-style discrete random variables questions in the AA SL question bank (Paper 1: 32 · Paper 2: 18), graded 14 easy, 20 medium, 11 hard, 5 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is discrete random variables on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 32 · Paper 2: 18. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA 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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Teaching this topic?

Free ready-to-teach lessons (slides and teacher notes): Discrete random variables & E(X) · Binomial distribution · Variance and standard deviation of X (extension).

IB Maths AA SL Discrete Random Variables Questions

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What you need to know

What's examined in AA SL discrete random variables

Key formulas

Where students lose marks on discrete random variables

Discrete Random Variables worked examples

Worked example 1: Finding probability constants and expected value · easy

Solution

Worked example 2: Solving for a missing prize in a fair game · medium

Solution

Worked example 3: Conditional binomial probability · hard

Solution

Try these IB Maths AA SL discrete random variables questions

Question 1 · easy · 4 marks · Paper 2

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 5 marks · Paper 2

Revise discrete random variables for the exam

FAQ

What else is free here

Teaching this topic?

More AA SL Unit 4 topics