FAD1015: Mathematics III — Tutorial 4

Centre for Foundation Studies in Science
Universiti Malaya
Session 2025/2026

Topic: Conditional Probability

Question 1

Given A and B are two events with $P(A) = 0.4$ and $P(B|A) = 0.7$ and $P(A' \cap B) = 0.3$. Find the following probabilities:

(a) $P(A \text{ and } B)$

(b) $P(B)$

(d) $P(A|B)$

(e) $P(B|A)$

Is $P(A|B) = P(B|A)$?

Question 2

A survey on the monthly income of the ex-students was carried out and the following table shows the results of a random sample of 1000 ex-students.

	Income per month in thousands (RM)
	< 4	[4, 10]	> 10
Class of 1990	250	300	80
Class of 1989	150	120	100

Find the probability that a randomly chosen ex student:

(a) was from Class of 1990 and earning less than RM4000,

(b) earns more than RM 10000 per month,

(d) was from Class of 1990 knowing that he earns more than RM4000 monthly.

Question 3

An urn has three blue balls and two white balls. If two balls are chosen at random one by one without replacement, find the probability that:

(a) the second ball is blue given that the first ball is white,

(b) both balls are blue.

Topic: Independent Events

Question 1

The probability that Ameera, Bella and Celline gets a grade A for their Mathematics subject in the mid semester examinations are $\frac{3}{4}$, $\frac{1}{2}$ and $\frac{4}{5}$ respectively. Since they are not able to copy from each other in the Examination Hall, we then assume that the event of getting an A is independent. Find the probability that:

(a) all three of them get an A,

(b) none of them gets an A,

Question 2

Box A has 4 red balls and 2 white balls. Box B has 2 red, 2 white and one green ball. In a game, a dice is thrown first and if the outcome is a number less than 3, a ball is chosen at random from Box A. Otherwise, a ball is randomly chosen from Box B.

Draw a Tree Diagram to show all the possible outcomes of the game. Hence, find the probability that:

(a) a red ball is chosen from Box A,

(b) a white ball is chosen from Box B,

(d) a ball is chosen from Box B if it is known that it is red.

Question 3

A college has 100 lecturers. In a survey on the use of the school car park, the lecturers were asked whether they had driven a car to college on a particular day. Of the 70 full-time lecturers, 45 had driven a car to college and of the 30 part-time lecturers, 12 had driven a car to college.

(a) Find the probability that a lecturer chosen at random:

i. is a part-time lecturer who had driven a car to college,

ii. is a full-time lecturer who had not driven a car to college,

iii. is a full-time lecturer or had driven a car to college,

iv. is a part-time lecturer, given that the lecturer had driven a car to college.

(b) Are the events 'the lecture had driven a car to college' and 'the lecturer is full-time' independent? Give a reason for your answer.

Topic: Bayes Theorem

Question 4

A computer company XX subcontracts the manufacturing of its monitor to two manufacturers, i.e 35% to A and 65% to B. Manufacturer A in turn subcontracts 60% of the orders from XX to manufacturer C and the remaining to D. Manufacturer B also subcontracts 30%, 50% and 20% of the orders from XX to E, F and G respectively. It was found that 1.5%, 1.0%, 1.2%, 1.25% and 0.85% of the monitors from C, D, E, F and G become defective within the 1-year warranty period from the date of purchase. What is the probability that:

(a) the monitor is defective during the 1 year warranty period?

(b) the monitor is from manufacturer C knowing that it is defective during the 1 year warranty period?

Question 5

Raqib drive to UM each day either using route A, B or C. The probabilities that he chooses route A and B are $\frac{1}{3}$ and $\frac{1}{5}$ respectively. The probability that he will be late for work using route A, B and C are $\frac{1}{5}$, $\frac{1}{4}$ and $\frac{3}{7}$ respectively.

(a) What is the probability that he will be late?

(b) If he is late, what is the probability that he uses route A?

Question 6

There are 90 applicants for a job with the project management department of an Engineering firm. Some of them are university graduates and some are not, some of them have at least 3 years working experience and some have less.

The numbers are shown in the following table:

	At least 3 years experience	Less than 3 years experience
University graduates	18	36
Non university graduates	9	27

Let $A$ be the event that the applicant interviewed is a university graduate and $B$ is the event that the applicant interviewed has at least 3 years experience.

(a) Find the following probabilities:

i. $P(A'|B')$

ii. $P(B|A)$

iii. $P(A \cup B)'$

(b) Are $A$ and $B$ independent events?

Related Concepts

Conditional Probability — probability of event A given that B has occurred
Bayes' Theorem — relating conditional probabilities
Independent Events — events where occurrence of one doesn't affect the other
Tree Diagram — visual representation of sequential probabilities
Total Probability — law of total probability

Source: FAD1015 Questions T1-T6 _20252026.pdf