ISYE 6644 OH Exam 1 Practice Questions – Spring 2025 Complete Study Guide with Verified Questions, Answers & Rationales.

ISYE 6644 OH Exam 1 Practice Questions – Spring 2025
Complete Study Guide with Verified Questions, Answers
& Rationales.


SECTION 1: PROBABILITY & STATISTICS FOUNDATIONS
(Questions 1-30)




Question 1
What is the probability of rolling a sum of 7 with two fair six-sided dice?

A) 1/36
B) 5/36
C) 6/36 = 1/6
D) 1/2

Correct Answer: C

Rationale: There are 6 combinations that sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
out of 36 total possible outcomes. 6/36 simplifies to 1/6.




Question 2
If P(A) = 0.3, P(B) = 0.4, and A and B are independent, what is P(A ∪ B)?

A) 0.12
B) 0.58
C) 0.70
D) 0.82

Correct Answer: B

Rationale: For independent events, P(A ∩ B) = P(A)P(B) = 0.3 × 0.4 = 0.12. Then P(A ∪ B)
= P(A) + P(B) − P(A ∩ B) = 0.3 + 0.4 − 0.12 = 0.58.

,Question 3
What is the value of the standard normal PDF φ(z) at z = 0?

A) 0
B) 0.3989
C) 0.5000
D) 1.0000

Correct Answer: B

Rationale: φ(0) = 1/√(2π) ≈ 0.3989. This is the maximum value of the standard normal
probability density function.




Question 4
For a standard normal random variable Z, what is P(Z > 1.96)?

A) 0.025
B) 0.05
C) 0.95
D) 0.975

Correct Answer: A

Rationale: From the standard normal table, P(Z > 1.96) = 0.025. This is a critical value
commonly used for 95% confidence intervals.




Question 5
What is the expected value of a Bernoulli(p) random variable?

A) 0
B) p
C) 1−p
D) 1

Correct Answer: B

,Rationale: For a Bernoulli(p) random variable X, P(X=1) = p and P(X=0) = 1−p. So E[X] =
1×p + 0×(1−p) = p.




Question 6
What is the variance of a Bernoulli(p) random variable?

A) p
B) 1−p
C) p(1−p)
D) p²

Correct Answer: C

Rationale: For a Bernoulli(p) random variable, Var(X) = p(1−p). This is the standard
formula for the variance of a Bernoulli distribution.




Question 7
A family has 2 children. Assume P(kid is a boy) = 1/3 and P(kid is a girl) = 2/3, and the
genders are independent. What is the probability that both kids are girls given that we
know at least 1 of them is a girl?

A) 1/16
B) 1/8
C) 1/5
D) 1/4
E) 1/2

Correct Answer: E

Rationale: The sample space is S = {GG, GB, BG, BB}. Let A = both are girls = {GG}; let
B = at least one girl = {GG, GB, BG}. P(GG) = (2/3)(2/3) = 4/9. P(at least one girl) = 1 −
P(BB) = 1 − (1/3)(1/3) = 8/9. P(GG | at least one girl) = (4/9)/(8/9) = 1/2.




Question 8

, TRUE or FALSE: The function f(x) = 5e⁻⁵ˣ for x ≥ 0 is a valid probability density function.

Correct Answer: True

Rationale: ∫₀^∞ 5e⁻⁵ˣ dx = 1. This is the exponential distribution with rate λ = 5.




Question 9
TRUE or FALSE: Discrete-event simulations are particularly suitable for analyzing
continuous-flow phenomena such as weather patterns propagating across the country.

Correct Answer: False

Rationale: Discrete-event simulations are not suitable for continuous-flow phenomena.
They are designed for systems where changes occur at discrete points in time (e.g.,
arrivals, departures). Continuous systems like weather patterns require continuous
simulation approaches.




Question 10
If X ~ Bernoulli(0.2), find E[4^X].

A) 0.8
B) 1.0
C) 1.6
D) 2.0

Correct Answer: C

Rationale: E[4^X] = 4^0·P(X=0) + 4^1·P(X=1) = 1·(0.8) + 4·(0.2) = 0.8 + 0.8 = 1.6.




Question 11
Find E[4X − 5] if E[X] = 6.

A) 12
B) 19