Probability Theory & Distributions Exam 2026: Comprehensive
University/Graduate Practice Examination — Probability Rules,
Random Variables, Conditional Probability, Bayes’ Theorem,
Discrete & Continuous Distributions, Expectation, Variance,
Sampling & Probability Models — 190+ Multiple-Choice Questions
with Answers & Detailed Rationales
SECTION 1: FOUNDATIONS OF PROBABILITY (Questions 1–25)
Question 1
Which term describes the set of all possible outcomes of a random experiment?
A. Event
B. Sample space
C. Random variable
D. Probability measure
Correct Answer: B
Explanation: The sample space S is the complete collection of all possible outcomes, forming
the foundation for defining events and probabilities.
Question 2
If P(A) = 0.4 and P(B) = 0.5, and A and B are mutually exclusive, what is P(A ∪ B)?
A. 0.1
B. 0.2
C. 0.9
D. 1.0
,Correct Answer: C
Explanation: For mutually exclusive events, P(A ∪ B) = P(A) + P(B) = 0.4 + 0.5 = 0.9, since no
overlap exists.
Question 3
A fair six-sided die is rolled. What is the probability of obtaining a prime number?
A. 1/6
B. 1/3
C. 1/2
D. 2/3
Correct Answer: C
Explanation: Primes on a die are 2, 3, and 5, giving 3/6 = 1/2 probability.
Question 4
Which axiom states that the probability of the entire sample space equals one?
A. First axiom
B. Second axiom
C. Third axiom
D. Complement axiom
Correct Answer: B
,Explanation: Kolmogorov's second axiom normalizes probability so that P(S) = 1 for the
whole sample space.
Question 5
If P(A) = 0.7, what is P(Aᶜ)?
A. 0.3
B. 0.7
C. 1.0
D. 0.0
Correct Answer: A
Explanation: The complement rule gives P(Aᶜ) = 1 − P(A) = 1 − 0.7 = 0.3.
Question 6
Two events A and B satisfy P(A ∩ B) = P(A)P(B). These events are:
A. Mutually exclusive
B. Exhaustive
C. Independent
D. Complementary
Correct Answer: C
Explanation: Statistical independence is defined by the factorization P(A ∩ B) = P(A)P(B).
Question 7
, A bag has 3 red and 5 blue marbles. One marble is drawn. What is P(red)?
A. 3/8
B. 5/8
C. 3/5
D. 1/3
Correct Answer: A
Explanation: Classical probability = favorable/total = 3/(3+5) = 3/8.
Question 8
Which rule computes P(A ∪ B) for non-mutually exclusive events?
A. P(A) + P(B)
B. P(A)P(B)
C. P(A) + P(B) − P(A ∩ B)
D. P(A) − P(B)
Correct Answer: C
Explanation: The inclusion–exclusion principle subtracts the double-counted intersection to
avoid overcounting.
Question 9
If P(A|B) = P(A), then events A and B are:
A. Dependent
University/Graduate Practice Examination — Probability Rules,
Random Variables, Conditional Probability, Bayes’ Theorem,
Discrete & Continuous Distributions, Expectation, Variance,
Sampling & Probability Models — 190+ Multiple-Choice Questions
with Answers & Detailed Rationales
SECTION 1: FOUNDATIONS OF PROBABILITY (Questions 1–25)
Question 1
Which term describes the set of all possible outcomes of a random experiment?
A. Event
B. Sample space
C. Random variable
D. Probability measure
Correct Answer: B
Explanation: The sample space S is the complete collection of all possible outcomes, forming
the foundation for defining events and probabilities.
Question 2
If P(A) = 0.4 and P(B) = 0.5, and A and B are mutually exclusive, what is P(A ∪ B)?
A. 0.1
B. 0.2
C. 0.9
D. 1.0
,Correct Answer: C
Explanation: For mutually exclusive events, P(A ∪ B) = P(A) + P(B) = 0.4 + 0.5 = 0.9, since no
overlap exists.
Question 3
A fair six-sided die is rolled. What is the probability of obtaining a prime number?
A. 1/6
B. 1/3
C. 1/2
D. 2/3
Correct Answer: C
Explanation: Primes on a die are 2, 3, and 5, giving 3/6 = 1/2 probability.
Question 4
Which axiom states that the probability of the entire sample space equals one?
A. First axiom
B. Second axiom
C. Third axiom
D. Complement axiom
Correct Answer: B
,Explanation: Kolmogorov's second axiom normalizes probability so that P(S) = 1 for the
whole sample space.
Question 5
If P(A) = 0.7, what is P(Aᶜ)?
A. 0.3
B. 0.7
C. 1.0
D. 0.0
Correct Answer: A
Explanation: The complement rule gives P(Aᶜ) = 1 − P(A) = 1 − 0.7 = 0.3.
Question 6
Two events A and B satisfy P(A ∩ B) = P(A)P(B). These events are:
A. Mutually exclusive
B. Exhaustive
C. Independent
D. Complementary
Correct Answer: C
Explanation: Statistical independence is defined by the factorization P(A ∩ B) = P(A)P(B).
Question 7
, A bag has 3 red and 5 blue marbles. One marble is drawn. What is P(red)?
A. 3/8
B. 5/8
C. 3/5
D. 1/3
Correct Answer: A
Explanation: Classical probability = favorable/total = 3/(3+5) = 3/8.
Question 8
Which rule computes P(A ∪ B) for non-mutually exclusive events?
A. P(A) + P(B)
B. P(A)P(B)
C. P(A) + P(B) − P(A ∩ B)
D. P(A) − P(B)
Correct Answer: C
Explanation: The inclusion–exclusion principle subtracts the double-counted intersection to
avoid overcounting.
Question 9
If P(A|B) = P(A), then events A and B are:
A. Dependent
