IB Maths AA HL · Unit 4: Statistics and Probability
IB Maths AA HL Conditional Probability Questions
Exam-style IB Maths AA HL conditional probability questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 28 questions
- Paper 1: 15
- Paper 2: 13
- 5 easy
- 7 medium
- 12 hard
- 4 starter
- 3 worked examples
Practise Conditional Probability questions →
AA HL formula booklet
What's examined in AA HL conditional probability
The question bank covers these conditional probability question types (number of questions in brackets):
- Event Algebra & Conditional Formula (12)
- Total Probability via Tree Diagrams (11)
- Bayes' Theorem & Inference (5)
Conditional Probability worked examples
Worked example 1: Finding conditional probability using the addition rule · easy
For two events $A$ and $B$, it is given that $P(A) = 0.45$, $P(B) = 0.60$, and $P(A \cup B) = 0.85$. Find the exact value of $P(A | B)$.
1. Identify the addition rule for probabilities: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
2. Substitute the known values into the equation to find the intersection: $0.85 = 0.45 + 0.60 - P(A \cap B)$.
3. Solve for the probability of the intersection: $P(A \cap B) = 1.05 - 0.85 = 0.20$.
4. State the formal definition of conditional probability: $P(A | B) = \frac{P(A \cap B)}{P(B)}$.
5. Substitute the calculated intersection and $P(B)$ into the conditional formula: $\frac{0.20}{0.60}$.
6. Simplify the fraction to find the exact probability: $\mathbf{\frac{1}{3}}$.
Examiner tip: Students often confuse $P(A|B)$ with $P(B|A)$ and consequently mistakenly divide the intersection probability by $P(A)$ instead of $P(B)$ when substituting into the conditional formula.
Worked example 2: Applying Bayes' theorem to reverse conditional probabilities · medium
A factory produces components using two machines. Machine A produces $60\%$ of the components and Machine B produces $40\%$. It is known that $3\%$ of the components from Machine A are defective, while $5\%$ of those from Machine B are defective. Given that a randomly selected component is defective, find the probability it was produced by Machine A. Give your answer to 3 significant figures.
1. Define the initial probabilities for the machines: $P(A) = 0.60$ and $P(B) = 0.40$.
2. Define the conditional probabilities for defects ($D$): $P(D|A) = 0.03$ and $P(D|B) = 0.05$.
3. Calculate the total probability of selecting a defective component using the law of total probability: $P(D) = P(A)P(D|A) + P(B)P(D|B)$.
4. Evaluate this total denominator probability: $P(D) = (0.60 \times 0.03) + (0.40 \times 0.05) = 0.018 + 0.020 = 0.038$.
5. Apply Bayes' theorem to find the reverse conditional probability $P(A|D)$: $P(A|D) = \frac{P(A \cap D)}{P(D)} = \frac{P(A)P(D|A)}{P(D)}$.
6. Substitute the values and simplify: $\frac{0.018}{0.038} = \frac{18}{38} = \frac{9}{19} \approx \mathbf{0.474}$.
Examiner tip: A classic error in Bayes' theorem questions is forgetting to include all corresponding branches of the tree diagram when calculating the denominator (the total probability of the condition being true).
Worked example 3: Solving for probabilities with independent events · hard
Events $C$ and $D$ are independent. It is given that $P(C) = 3P(D)$ and $P(C \cup D) = 0.68$. Find the exact value of $P(D)$.
1. Write the general addition rule for probabilities: $P(C \cup D) = P(C) + P(D) - P(C \cap D)$.
2. Substitute the product of probabilities for the intersection, since the events are explicitly independent: $P(C \cap D) = P(C)P(D)$.
3. Set up the full algebraic equation by letting $x = P(D)$, meaning $P(C) = 3x$: $0.68 = 3x + x - (3x)(x)$.
4. Rearrange the terms to form a standard quadratic equation: $3x^2 - 4x + 0.68 = 0$.
5. Solve the quadratic equation by multiplying by $100$ and factorising (or using the quadratic formula): $300x^2 - 400x + 68 = 0 \implies 75x^2 - 100x + 17 = 0 \implies x = 0.2$ or $x = \frac{17}{15}$.
6. Reject the root that is strictly greater than $1$, giving the final exact probability: $\mathbf{P(D) = 0.2}$.
Examiner tip: Students often try to solve this by incorrectly assuming mutually exclusive events and writing $P(C \cup D) = P(C) + P(D)$, completely ignoring the crucial intersection term generated by independence.
Try these IB Maths AA HL conditional probability questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 4 marks · Paper 1
Events \(A\) and \(B\) are represented in a universal set. It is given that \(P(A) = 0.4\), \(P(A|B) = 0.25\), and \(P(A \cup B) = 0.55\).
Find the exact value of \(P(B)\), and hence find \(P(B|A)\).
Question 2 · medium · 6 marks · Paper 1
In a team of 30 judo players, 13 have won a match by throwing (\(T\)), 12 have won by hold-down (\(H\)), and 13 have won by points decision (\(P\)).
2 players have won matches by all three methods. 5 have won matches by throwing and hold-down. 4 have won matches by hold-down and points decision. 3 have won matches by throwing and points decision.
Find the exact probability that a randomly selected player has won a match by throwing (\(T\)), given that they have won a match by exactly one method.
Question 3 · hard · 6 marks · Paper 2
At a second pet shop, let the events \(G\) and \(T\) represent buying a goldfish and a tortoise respectively.
The probability of buying a tortoise is twice the probability of buying a goldfish: \(P(T) = 2P(G)\).
It is known that \(G\) and \(T\) are independent events, and the probability of buying neither is \(0.28\).
Find \(P(G|T \cup G)\), the probability a customer bought a goldfish given that they bought at least one of the two pets.
All 28 conditional probability questions with mark schemes →
FAQ
How many IB Maths AA HL conditional probability questions are there?
There are 28 exam-style conditional probability questions in the AA HL question bank (Paper 1: 15 · Paper 2: 13), graded 5 easy, 7 medium, 12 hard, 4 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is conditional probability on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 15 · Paper 2: 13. 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 HL 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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