Two fair 5-sided dice, each numbered 1 to 5, are rolled. The sum of the two numbers shown is recorded. Find the probability that the sum of the two dice is strictly greater than 7. Given that the sum of the two dice is strictly greater than 7, find the probability that the sum is exactly 9.

Total number of equally likely outcomes $= 5 \times 5 = 25$ . Sums strictly greater than 7 are 8, 9 and 10. (M1) Sum 8: (3,5), (4,4), (5,3); sum 9: (4,5), (5,4); sum 10: (5,5). Total $= 6$ outcomes. (A1) $P(\text{Sum} > 7) = \frac{6}{25} = 0.24$ . A1 The reduced sample space is the 6 outcomes with sum $>7$ . (M1) Of these, 2 have a sum of exactly 9. Probability $= \frac{2}{6} = \frac{1}{3} \approx 0.333$ . A1

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IB Maths AI SL · Unit 4: Statistics and Probability

IB Maths AI SL Probability Questions

Updated October 2026

Exam-style IB Maths AI SL 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.

60 questions
Paper 1: 36
Paper 2: 24
13 easy
18 medium
20 hard
9 very hard
3 worked examples

Practise 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 probability

The question bank covers these probability question types (number of questions in brackets):

Conditional & Compound Events (35)
Venn Diagrams & Set Formulas (22)
Sample Space Enumeration (3)

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 →

Where students lose marks on probability

Examiner's noteProbability: adding instead of multiplying, conditional sample spaces, without replacement

Where marks are lost: Candidates added probabilities that should be multiplied (and doubled a probability that should be squared), gave probabilities greater than 1, treated conditional events as independent instead of restricting the sample space, and treated selections without replacement as binomial. Tree diagrams and tables were rarely drawn.

What examiners look for: Draw the tree or table. Along a branch multiply; between outcomes add. For "given that", the denominator is the size of the given group. Without replacement, the second probability changes.

Examiners flagged this in 3 exam series (May 2024, November 2024, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Probability worked examples

Worked example 1: Sample space and simple probability · easy

A fair coin and a fair 4-sided die (1–4) are rolled. Find the exact probability of a Head and a prime number.

Solution

5 steps

Examiner tip: Listing the full sample space visually (grid/table) prevents missing overlapping outcomes under exam pressure.

Worked example 2: Tree diagram — exactly one of two days · medium

$P (rain) = 0.4$ each day, independent. Find the exact probability of rain on exactly one of two consecutive days.

Solution

5 steps · 1 to fill in

Examiner tip: "Exactly one day" can occur in two chronological orders — both must be added.

Worked example 3: Conditional probability without replacement · hard

Bag: $4$ red, $5$ green marbles. Two drawn without replacement. Given the second is green, find $P (first is green)$ .

Solution

5 steps · 1 to fill in

Examiner tip: "Without replacement" — the denominator drops from 9 to 8, and the chosen colour's numerator drops by 1.

Try these IB Maths AI SL 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 · easy · 4 marks · Paper 1

Events A and B are mutually exclusive. It is given that $P (A) = 0.25$ and $P (B) = 0.35$ .

Find $P (A \cup B)$ .
Find $P (A^{'} \cap B^{'})$ , the probability that neither event A nor event B occurs.

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

Two fair 5-sided dice, each numbered 1 to 5, are rolled. The sum of the two numbers shown is recorded.

Find the probability that the sum of the two dice is strictly greater than 7.
Given that the sum of the two dice is strictly greater than 7, find the probability that the sum is exactly 9.

Full worked solution Free, no sign-in

Total number of equally likely outcomes $= 5 \times 5 = 25$ .
Sums strictly greater than 7 are 8, 9 and 10. (M1)
Sum 8: (3,5), (4,4), (5,3); sum 9: (4,5), (5,4); sum 10: (5,5). Total $= 6$ outcomes. (A1)
$P (Sum > 7) = \frac{6}{25} = 0.24$ . A1
The reduced sample space is the 6 outcomes with sum $> 7$ . (M1)
Of these, 2 have a sum of exactly 9.
Probability $= \frac{2}{6} = \frac{1}{3} \approx 0.333$ . A1

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

Question 3 · hard · 4 marks · Paper 1

Events A and B are mutually exclusive. It is known that $P (A) = 2 x$ and $P (B) = x$ . Given that $P (A \cup B) = 0.9$ , find the value of $x$ .

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AI SL probability questions are there?

There are 60 exam-style probability questions in the AI SL question bank (Paper 1: 36 · Paper 2: 24), graded 13 easy, 18 medium, 20 hard, 9 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is probability on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 36 · Paper 2: 24. 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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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): Combined events & sample-space diagrams · Conditional probability · Discrete distributions & E(X).

Probability in other IB Maths courses

IB Maths AA SL Probability questions

Keep revising AI SL: AI SL revision notes · IB Maths AI SL past papers · AI SL common exam mistakes · AI SL syllabus checklist · AI SL statistics and probability knowledge organiser

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IB Maths AI SL Probability Questions

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What you need to know

What's examined in AI SL probability

Key formulas

Where students lose marks on probability

Probability worked examples

Worked example 1: Sample space and simple probability · easy

Solution

Worked example 2: Tree diagram — exactly one of two days · medium

Solution

Worked example 3: Conditional probability without replacement · hard

Solution

Try these IB Maths AI SL probability questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 4 marks · Paper 1

Revise probability for the exam

FAQ

What else is free here

Teaching this topic?

More AI SL Unit 4 topics

Probability in other IB Maths courses