Bayes' theorem: IB Mathematics: Analysis & Approaches questions with answers
Free Bayes' theorem questions for IB Mathematics: Analysis & Approaches (SL & HL), HL only, part of Statistics & probability. Each question has the answer and the marking points an examiner would look for.
A disease affects 2% of a population. A test gives a positive result for 95% of people with the disease and for 5% of people without it.
(a)
Find the probability that a randomly chosen person tests positive.
[2]
(b)
Given that a person tests positive, find the probability that they have the disease.
[3]
(c)
Comment on the usefulness of the test.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
17/250 (= 0.068)
tree or total probability (M1) 0.068 = 17/250 A1
2
(b)
19/68 (≈ 0.279)
conditional probability (M1) 0.0190.068(A1) 19/68 (≈ 0.279) A1
3
(c)
many false positives
most positive results are false positives, so a positive result alone is not reliable R1
1
Question 2Structured · 6 marks
Machines A, B and C produce 40%, 35% and 25% of the items made in a factory. The percentages of defective items they produce are 3%, 2% and 4% respectively. An item is chosen at random.
(a)
Find the probability that the item is defective.
[3]
(b)
Given that the item is defective, find the probability that it was made by machine B.
A disease affects 1% of a population. A test gives a positive result for 95% of people with the disease and for 5% of people without it.
(a)
Find the probability that a randomly chosen person tests positive.
[2]
(b)
Given that a person tests positive, find the probability that they have the disease.
[3]
(c)
Comment on the usefulness of the test.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
59/1000 (= 0.059)
tree or total probability (M1) 0.059 = 59/1000 A1
2
(b)
19/118 (≈ 0.161)
conditional probability (M1) 0.00950.059(A1) 19/118 (≈ 0.161) A1
3
(c)
many false positives
most positive results are false positives, so a positive result alone is not reliable R1
1
Question 4Structured · 6 marks
Machines A, B and C produce 40%, 35% and 25% of the items made in a factory. The percentages of defective items they produce are 2%, 5% and 4% respectively. An item is chosen at random.
(a)
Find the probability that the item is defective.
[3]
(b)
Given that the item is defective, find the probability that it was made by machine C.
A disease affects 5% of a population. A test gives a positive result for 95% of people with the disease and for 10% of people without it.
(a)
Find the probability that a randomly chosen person tests positive.
[2]
(b)
Given that a person tests positive, find the probability that they have the disease.
[3]
(c)
Comment on the usefulness of the test.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
57/400 (= 0.1425)
tree or total probability (M1) 0.1425 = 57/400 A1
2
(b)
1/3 (≈ 0.333)
conditional probability (M1) 0.04750.1425(A1) 1/3 (≈ 0.333) A1
3
(c)
many false positives
most positive results are false positives, so a positive result alone is not reliable R1
1
Question 6Structured · 6 marks
Machines A, B and C produce 40%, 35% and 25% of the items made in a factory. The percentages of defective items they produce are 1%, 4% and 6% respectively. An item is chosen at random.
(a)
Find the probability that the item is defective.
[3]
(b)
Given that the item is defective, find the probability that it was made by machine A.
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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in