Practice SL 4.5—Probability concepts, expected numbers with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.
Paper
Level
Question 1
SL & HLPaper 1
A bag contains 12 cards labeled with the numbers 1 to 12 . Three cards are drawn randomly without replacement.
1.[1]
Find the probability that the first card drawn is even-numbered.
2.[2]
Find the probability that the second card drawn is odd-numbered given that the first card was even-numbered.
3.[3]
Find the probability that all three cards drawn are even-numbered.
Question 2
SL & HLPaper 1
A survey of 60 students at a school shows that 32 take biology (B), 28 take chemistry (C), and 15 take both subjects.
1.[2]
Complete the Venn diagram to represent the subject choices.
2.[1]
Find the number of students who take neither biology nor chemistry.
3.[2]
A student is chosen at random. Find the probability that the student takes chemistry given that they take biology.
4.[2]
Find the expected number of students taking only biology in a random sample of 20 students.
Question 3
SL & HLPaper 2
A school organizes a survey of 120 students participating in three clubs: literature (L), science (S), and drama (D). The following information is known:
- 50 students are in the literature club, 60 are in the science club, and 45 are in the drama club.
- 20 students are in both literature and science, 15 are in both literature and drama, and 25 are in both science and drama.
- 10 students are in all three clubs.
- 12 students are not in any of the clubs.
A game is played where a student is selected at random, and points are awarded based on club membership: 3 points for each club they are in, but if they are in no clubs, they lose 5 points. Let represent the points earned.
1.[5]
Draw a Venn diagram to represent the club memberships, labeling sets L, S, and D.
2.[3]
Find the probability that a randomly selected student is in exactly two clubs.
3.[4]
Construct the probability distribution table for .
4.[3]
Calculate the expected value of .
5.[3]
Determine whether membership in the literature and science clubs is independent.
Question 4
SL & HLPaper 2
A store sells three types of drinks: cola (C), juice (J), and water (W). A survey of 80 customers shows that 45 buy cola, 40 buy juice, 30 buy water, 20 buy both cola and juice, 15 buy both cola and water, 10 buy both juice and water, and 5 buy all three.
1.[2]
Complete the Venn diagram for drink purchases.
2.[2]
Find the probability that a randomly chosen customer buys at least two types of drinks.
3.[2]
Find the expected number of customers buying only juice in a sample of 25 customers.
Question 5
SL & HLPaper 2
A box contains 10 marbles: 4 red (R), 3 blue (B), and 3 green (G). Two marbles are drawn randomly without replacement.
1.[1]
Find the probability that the first marble drawn is red.
2.[2]
Find the probability that the second marble drawn is blue given that the first marble was green.
3.[3]
Find the probability that both marbles drawn are of the same color.
Question 6
SL & HLPaper 2
A school organizes two extracurricular clubs: debate (D) and chess (C). Out of 150 students, the probability that a randomly chosen student is in the debate club is , and the probability that they are in both clubs is . There are 25 students who are not in either club.
1.[2]
Complete the Venn diagram to represent the club memberships.
2.[2]
Find the probability that a randomly chosen student who is in the chess club is also in the debate club.
3.[2]
Determine whether the events of being in the debate club and the chess club are independent.
Question 7
SL & HLPaper 1
A community center offers three activities: yoga (Y), painting (P), and cooking (C). A survey of 100 members showed the following preferences:
- 40 members liked yoga, 35 liked painting, and 30 liked cooking.
- 5 members liked all three activities.
- 15 members liked both yoga and painting.
- 10 members liked both yoga and cooking.
- 8 members liked both painting and cooking.
1.[2]
Complete the following Venn diagram to represent the survey results.
2.[2]
Find the number of members who did not like any of the three activities.
3.[2]
A member is chosen at random. Find the probability that this member likes cooking given that they like painting.
Question 8
SL & HLPaper 2
A club has 100 members, with 60 participating in hiking and 50 in cycling . The probability that a member participates in both is .
1.[2]
Complete the Venn diagram for the activities.
2.[2]
Find the probability that a randomly chosen member participates in hiking but not cycling.
3.[2]
Find the expected number of members participating in at least one activity in a sample of 40 members.
4.[2]
Determine whether participating in hiking and cycling are independent events.
Question 9
SL & HLPaper 2
A library tracks borrowing of fiction (F) and non-fiction (N) books by 200 patrons. The probability a patron borrows fiction is , and 80 patrons borrow both types.
1.[2]
Complete the Venn diagram for book borrowing.
2.[2]
Find the expected number of patrons borrowing only non-fiction in a sample of 50 patrons.
3.[2]
Determine whether borrowing fiction and non-fiction are independent events.
4.[2]
A patron who borrows non-fiction is chosen at random. Find the probability they also borrow fiction.
Question 10
SL & HLPaper 1
In a group of 50 employees, 30 attend a morning meeting (M), 25 attend an evening meeting (E), and 10 attend both.
1.[2]
Complete the Venn diagram to represent the meeting attendance.
2.[2]
A randomly chosen employee attends the evening meeting. Find the probability that they also attend the morning meeting.
3.[2]
Determine whether attending the morning and evening meetings are independent events.