ISYE 6644 PROBABILITY AND STATISTICS FOR
ENGINEERING 2026: COMPREHENSIVE MIDTERM &
FINAL EXAM STUDY GUIDE | VERIFIED QUESTIONS,
ANSWERS, DETAILED SOLUTIONS & COMPLETE
EXAM PREPARATION
SECTION 1: PROBABILITY THEORY AND AXIOMS (QUESTIONS 1-10)
1. Which of the following represents the correct set of axioms for a probability function
P defined on a sample space S?
A. P(A) >= 0, P(S) = 0, For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B)
B. P(A) <= 1, P(S) = 1, For mutually exclusive events A and B, P(A ∩ B) = P(A)P(B)
C. P(A) >= 0, P(S) = 1, For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B)
D. P(A) > 0, P(S) = 1, For any events A and B, P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
2. Let A and B be two events with P(A) = 0.6, P(B) = 0.4, and P(A ∩ B) = 0.2. What is the
probability of the complement of (A ∪ B)?
A. 0.2
B. 0.4
C. 0.6
D. 0.8
3. For two independent events A and B, if P(A) = 0.3 and P(B) = 0.5, what is P(A ∪ B)?
A. 0.15
B. 0.65
C. 0.8
D. 0.35
, 4. A certain system can experience three different types of failures, labeled I, II, and III.
The probabilities of these failures are P(I)=0.1, P(II)=0.2, and P(III)=0.3. The
probability of exactly one of these failures occurring is 0.4. What is the probability
that none of these failures occur?
A. 0.4
B. 0.6
C. 0.3
D. 0.5
5. If A and B are mutually exclusive events, then which of the following statements is
always true?
A. P(A ∩ B) = P(A)P(B)
B. P(A ∪ B) = P(A) + P(B)
C. P(A | B) = P(A)
D. A and B are independent
6. The probability of event A is 0.4 and the probability of event B is 0.7. The minimum
possible value of P(A ∪ B) is:
A. 0.7
B. 0.4
C. 1.0
D. 0.28
7. Given that P(A) = 1/3, P(B) = 1/4, and P(A ∩ B) = 1/8, find P(A^c ∪ B^c).
A. 7/8
B. 1/8
C. 5/8
D. 3/4
8. For two events A and B, if P(A) = 0.5, P(B) = 0.4, and P(A | B) = 0.6, what is P(A ∩ B)?
A. 0.24
B. 0.20
C. 0.30
D. 0.15
9. Which of the following is a valid probability distribution for a discrete random
variable X with values 1, 2, and 3?
A. P(X=1)=0.2, P(X=2)=0.5, P(X=3)=0.3
B. P(X=1)=0.5, P(X=2)=0.5, P(X=3)=0.5
ENGINEERING 2026: COMPREHENSIVE MIDTERM &
FINAL EXAM STUDY GUIDE | VERIFIED QUESTIONS,
ANSWERS, DETAILED SOLUTIONS & COMPLETE
EXAM PREPARATION
SECTION 1: PROBABILITY THEORY AND AXIOMS (QUESTIONS 1-10)
1. Which of the following represents the correct set of axioms for a probability function
P defined on a sample space S?
A. P(A) >= 0, P(S) = 0, For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B)
B. P(A) <= 1, P(S) = 1, For mutually exclusive events A and B, P(A ∩ B) = P(A)P(B)
C. P(A) >= 0, P(S) = 1, For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B)
D. P(A) > 0, P(S) = 1, For any events A and B, P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
2. Let A and B be two events with P(A) = 0.6, P(B) = 0.4, and P(A ∩ B) = 0.2. What is the
probability of the complement of (A ∪ B)?
A. 0.2
B. 0.4
C. 0.6
D. 0.8
3. For two independent events A and B, if P(A) = 0.3 and P(B) = 0.5, what is P(A ∪ B)?
A. 0.15
B. 0.65
C. 0.8
D. 0.35
, 4. A certain system can experience three different types of failures, labeled I, II, and III.
The probabilities of these failures are P(I)=0.1, P(II)=0.2, and P(III)=0.3. The
probability of exactly one of these failures occurring is 0.4. What is the probability
that none of these failures occur?
A. 0.4
B. 0.6
C. 0.3
D. 0.5
5. If A and B are mutually exclusive events, then which of the following statements is
always true?
A. P(A ∩ B) = P(A)P(B)
B. P(A ∪ B) = P(A) + P(B)
C. P(A | B) = P(A)
D. A and B are independent
6. The probability of event A is 0.4 and the probability of event B is 0.7. The minimum
possible value of P(A ∪ B) is:
A. 0.7
B. 0.4
C. 1.0
D. 0.28
7. Given that P(A) = 1/3, P(B) = 1/4, and P(A ∩ B) = 1/8, find P(A^c ∪ B^c).
A. 7/8
B. 1/8
C. 5/8
D. 3/4
8. For two events A and B, if P(A) = 0.5, P(B) = 0.4, and P(A | B) = 0.6, what is P(A ∩ B)?
A. 0.24
B. 0.20
C. 0.30
D. 0.15
9. Which of the following is a valid probability distribution for a discrete random
variable X with values 1, 2, and 3?
A. P(X=1)=0.2, P(X=2)=0.5, P(X=3)=0.3
B. P(X=1)=0.5, P(X=2)=0.5, P(X=3)=0.5
