Question: BLUECREST UNIVERSITY
Monrovia, Liberia
SCHOOL OF GRADUATE STUDIES
SPRING 2026
QUESTION PAPER
Name of the Program: MSc IT
Name of the Subject: Advanced Mathematics for
IT
Subject Code: MIT556
Semester: 1
Date of the Exam: 14 August, 2026
Exam-type: Mid-term (Resit)
Student Name:
Duration of the Exam: 1 Hour 30 Minutes
Total Marks: 30
Section A [Short Answer]
Answer any 5 questions from section A. Each question carries 2-marks. (5x2 = 10 Marks)
1. Define biconditional proposition in logic with example.
2. Express the statement If it rains, then the roads are wet in symbolic form.
3. Give the symbolic form of the statement All students who study regularly pass the
examination.
4. A box contains 8 red balls, 5 blue balls, and 3 green balls. Two balls are drawn at
random without replacement. Find the probability that both are red.
5. What is the probability that a non-leap year selected at random will contain 53
Sundays?
Section B [Long Answers]
Answer any 4 questions from section B. Each question carries 5-marks. (4 * 5 = 20 Marks)
1. Show that QV(P2Q)V(2PV2Q) is a tautology.
2. Show that (x) (P(x) 12 Q(x)) 2(Q(x) 12 R(x)) (x)(P(x) 12 R(x)).
3. Obtain the PDNF(Principal Disjunctive Normal Form) of (p 2 q) V (2p 2 r) V (q 2 r).
4. A bolt is manufactured by 3 machines A, B and C. A turns out twice as many items as
B, and machines B and C produce equal number of items. 2% of bolts produced by A
and B are defective and 4% of bolts produced by C are defective. All bolts are put into
1 stock pile and is chosen from this pile. What is the probability that it is defective?
5. A random variable X has the following probability distribution.
X -2 -1 0 1 2 3
P(X) 0.1 K 0.2 2k 0.3 3k
(a)Find K, (b) Evaluate P(X < 2) and P(-2 < X < 2) and (c) find the cdf of X
6. The probability that Azher, who always walks to work, arrives late on any day is 0.11.
Calculate the probability that, during a period of 20 working days, he is late on (i)
exactly 4 days; (ii) More than 2 days; and (iii) none of the day.
***All the Best***
1. **Problem:** Find the probability that both balls drawn without replacement from a box containing 8 red, 5 blue, and 3 green balls are red.
2. **Step 1: Understand the problem.**
- Total balls = $8 + 5 + 3 = 16$
- We draw 2 balls without replacement.
- We want the probability both are red.
3. **Step 2: Use the probability formula for dependent events:**
$$P(A \text{ and } B) = P(A) \times P(B|A)$$
where
- $P(A)$ is the probability the first ball is red.
- $P(B|A)$ is the probability the second ball is red given the first was red.
4. **Step 3: Calculate $P(A)$:**
$$P(A) = \frac{8}{16} = \frac{1}{2}$$
5. **Step 4: Calculate $P(B|A)$:**
After drawing one red ball, remaining red balls = $8 - 1 = 7$
Total remaining balls = $16 - 1 = 15$
$$P(B|A) = \frac{7}{15}$$
6. **Step 5: Calculate the combined probability:**
$$P(\text{both red}) = \frac{1}{2} \times \frac{7}{15} = \frac{7}{30}$$
7. **Final answer:**
$$\boxed{\frac{7}{30}}$$