CAIE A-Level Mathematics 5.3 Probability Question Bank

CAIE A-Level Mathematics 5.3 Probability Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise building probability models from tables, trees and counting, then applying addition, multiplication, complements, conditional probability and independence tests.

Exam points

  • translate the outcome structure into a tree, table or counting model before assigning probabilities
  • add mutually exclusive routes and multiply successive probabilities, adjusting for replacement
  • use conditional fractions or compare P(A ∩ B) with P(A)P(B) to test independence

Covered syllabus points

5.3.1—Permutations and combinations5.3.2—Probability rules5.3.3—Exclusive and independent events5.3.4—Elementary probability

Question 1

[Maximum number: 2]

At a college, the students choose exactly one of tennis, hockey or netball to play. The table shows the numbers of students in Year 1 and Year 2 at the college playing each of these sports.

Table for Question 1 — CAIE A-Level Mathematics

One student is chosen at random from the 120 students. Events X and N are defined as follows:
X: the student is in Year 1
N: the student plays netball.

Question 1(b)

(a)

Find P(N∣X)\mathrm{P}(N \mid X).

[ 1 ]

Question 1(c)

(b)

Determine whether or not X and N are independent events.

[ 1 ]

Question 3

[Maximum number: 8]

Priti has two bags of discs, X and Y.
Bag X contains 8 red discs and 7 blue discs.
Bag Y contains 6 red discs and 9 blue discs.
Priti tosses a fair coin.
If she obtains a head, she chooses at random and without replacement two discs from bag X.
If she obtains a tail, she chooses at random and with replacement two discs from bag Y.

Question 3(a)

(a)

Draw a tree diagram to represent this information, showing all the probabilities.

Table for Question 3(a) — CAIE A-Level Mathematics
[ 2 ]

Question 3(b)

(b)

Find the probability that the two discs that Priti chooses are blue.

[ 2 ]

Question 3(c)

(c)

Find the probability that the two discs that Priti chooses come from bag X given that at least one of the discs is red.

[ 4 ]

Question 5

[Maximum number: 8]

Chen has three boxes. Box A contains 5 counters, of which 3 are white and 2 are yellow. Box W contains 4 red marbles and 3 blue marbles. Box Y contains 5 red marbles and 3 blue marbles.

Chen chooses one counter at random from box A. If the counter is white, he chooses two marbles from box W, at random and without replacement. If the counter is yellow, he chooses two marbles from box Y, at random and without replacement.

Question 5(a)

(a)

Draw a fully labelled tree diagram to illustrate this information, including all the probabilities.

[ 3 ]

Question 5(b)

(b)

Find the probability that one red marble and one blue marble are obtained.

[ 3 ]

Question 5(c)

(c)

Find the probability that a white counter is chosen from box A, given that one red marble and one blue marble are obtained.

[ 2 ]

Question 7(c)

[Maximum number: 3]

Two letters are chosen at random from the 10 letters in the word ZOOLOGICAL. Find the probability that these two letters are different.