DTSA 5001 — Probability Theory P ractice Exam 250 questions , correct answers and bold rationales| LATEST UPDATE

DTSA 5001 — Probability Theory Practice
Exam 250 questions , correct answers and bold
rationales| LATEST UPDATE
Topics covered include : Sample Spaces & Axioms , Conditional
Probability, Random Variables, Distributions, Expectation & Variance, Limit
Theorems & Applications
Q1
Sample Spaces & Axioms
A sample space S for flipping a fair coin twice is:
A) {H, T}
✓ B) {HH, HT, TH, TT} (correct answer)
C) {2H, 1H1T, 2T}
D) {0, 1, 2}
Rationale: The sample space lists every possible outcome of the
experiment. Flipping a coin twice gives four equally likely sequences: HH,
HT, TH, TT. Option A is the single-flip space; option C collapses order;
option D counts heads.
Q2
Sample Spaces & Axioms
Which of the following correctly states Kolmogorov's first axiom of
probability?
A) P(A) can be any real number
✓ B) P(A) ≥ 0 for every event A (correct answer)
C) P(A) + P(B) = 1 for any two events
D) P(S) = 0

,Rationale: Kolmogorov's first axiom requires non-negativity: P(A) ≥ 0. The
three axioms are: (1) non-negativity, (2) normalization P(S)=1, (3)
countable additivity for mutually exclusive events.
Q3
Sample Spaces & Axioms
If A and B are mutually exclusive events, then P(A ∪ B) equals:
A) P(A) · P(B)
B) P(A) + P(B) − P(A ∩ B)
✓ C) P(A) + P(B) (correct answer)
D) 1 − P(A) − P(B)
Rationale: Mutually exclusive events cannot occur simultaneously, so P(A
∩ B) = 0. The addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) simplifies to
P(A) + P(B).
Q4
Sample Spaces & Axioms
The complement rule states that P(Aᶜ) equals:
A) P(A) − 1
✓ B) 1 − P(A) (correct answer)
C) 1 / P(A)
D) P(A) + 1
Rationale: Since A and Aᶜ partition the sample space, P(A) + P(Aᶜ) = 1,
giving P(Aᶜ) = 1 − P(A). This is one of the most-used derived rules in
probability.
Q5
Sample Spaces & Axioms

,A fair six-sided die is rolled. What is the probability of rolling a number
greater than 4?
A) 1/6
✓ B) 1/3 (correct answer)
C) 1/2
D) 2/3
Rationale: Numbers greater than 4 are {5, 6} — two outcomes out of six
equally likely outcomes: P = 2/6 = 1/3.
Q6
Sample Spaces & Axioms
Which of the following best describes a sigma-algebra (σ-algebra)?
A) A collection of events closed under finite union only
✓ B) A collection of subsets of S closed under complementation and
countable unions, containing ∅ (correct answer)
C) The set of all outcomes in S
D) A probability function mapping events to [0,1]
Rationale: A σ-algebra F is a collection of subsets of S satisfying: (1) ∅ ∈
F, (2) closed under complementation, (3) closed under countable unions. It
defines the valid events to which probability can be assigned.
Q7
Sample Spaces & Axioms
For any two events A and B, the general addition rule is:
A) P(A ∪ B) = P(A) · P(B)
B) P(A ∪ B) = P(A) + P(B)
✓ C) P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (correct answer)
D) P(A ∪ B) = P(A) − P(B)

, Rationale: The inclusion-exclusion principle: P(A ∪ B) = P(A) + P(B) − P(A
∩ B). The intersection is subtracted because it is counted twice when the
individual probabilities are added.
Q8
Sample Spaces & Axioms
A bag contains 3 red and 5 blue marbles. If one marble is drawn uniformly
at random, P(red) is:
A) 3/5
B) 5/8
✓ C) 3/8 (correct answer)
D) 1/3
Rationale: Total marbles = 8; red marbles = 3; P(red) = 3/8. Classical
probability assigns equal weight to each of the 8 equally likely outcomes.
Q9
Sample Spaces & Axioms
If events A, B, and C are mutually exclusive and exhaustive, which must be
true?
A) P(A) = P(B) = P(C) = 1/3
✓ B) P(A) + P(B) + P(C) = 1 (correct answer)
C) P(A ∩ B) = 1
D) A = B = C
Rationale: Mutually exclusive and exhaustive means no overlap and
together they cover S, so their probabilities sum to 1. They need not be
equally probable.
Q10
Sample Spaces & Axioms