CAIE A-Level Mathematics A2 5.3 Probability Questions

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 combinations 5.3.2—Probability rules 5.3.3—Exclusive and independent events 5.3.4—Elementary probability

[Maximum number: 4]

The 9 letters of the word RECORDERS are divided at random into two groups: a group of 5 letters and a group of 4 letters.

Find the probability that the three Rs are in the same group.

The favourable cases are:

- Group of 5 contains the three R's and two E's: 1 way.
- Group of 5 contains the three R's and one E: $\binom21\binom41=8$ ways.
- Group of 5 contains the three R's and no E: $\binom42=6$ ways.
- Group of 4 contains the three R's and one E: $\binom21=2$ ways.
- Group of 4 contains the three R's and no E: $\binom41=4$ ways.

Thus the total number of favourable splits is 21. The total number of splits is $\binom95=126$ , so
$P(\text{three }R\text{'s in the same group})=\frac{21}{126}=\frac16=0.167.$

[Maximum number: 7]

Box A contains 6 green balls and 3 yellow balls.
Box B contains 4 green balls and x yellow balls.
A ball is chosen at random from box A and placed in box B. A ball is then chosen at random from box B.

(a)

Draw a tree diagram to represent this information, showing the probability on each of the branches.

[ 4 ]

(b)

The probability that both the balls chosen are the same colour is $\frac{8}{15}$ .

Find the value of x.

[ 3 ]

The probability of choosing two balls of the same colour is
$\frac69\left(\frac5{5+x}\right)+\frac39\left(\frac{1+x}{5+x}\right)=\frac8{15}.$
Rearranging,
15(x+11)=24(x+5),
so
x=5.

[Maximum number: 7]

Jasmine throws two ordinary fair 6-sided dice at the same time and notes the numbers on the uppermost faces. The events A and B are defined as follows.
A : The sum of the two numbers is less than 6 .
$B: \quad$ The difference between the two numbers is at most 2 .

(a)

Determine whether or not the events A and B are independent.

[ 4 ]

$P(A)=\frac{10}{36},\qquad P(B)=\frac{24}{36},\qquad P(A\cap B)=\frac8{36}.$
But
$P(A)P(B)=\frac{10}{36}\frac{24}{36}=\frac5{27}\ne\frac8{36}.$
Therefore, events A and B are not independent.

(b)

Find $\mathrm{P}\left(B \mid A^{\prime}\right)$ .

[ 3 ]

$P(B\mid A')=\frac{P(B\cap A')}{P(A')}=\frac{16/36}{1-10/36} =\frac{16}{26}=\frac8{13}.$

All question bank results loaded

Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)