CS 7638 Midterm 1 (Fall 2026) - Practice
TestCourse
Instructor: Based on the work of Sebastian Thrun
Total Questions: 85
Instructions: Choose the best possible answer for each question.
Section 1: Probability Fundamentals (Questions 1-15)
1. In probability, the product rule, P(A,B) = P(A|B)P(B), is also known as:
a) The Theorem of Total Probability
b) Bayes' Rule
c) Chain Rule [rationale: The chain rule describes how to calculate the probability of
the intersection of multiple events by multiplying conditional probabilities.]
d) The Law of Large Numbers
2. Which of the following statements is always true for any two random events
A and B?
a) P(A|B) = P(B|A)
b) P(A,B) = P(A) * P(B)
c) P(A ∪ B) = P(A) + P(B) - P(A,B) [rationale: This is the fundamental principle of
inclusion-exclusion for the union of two events.]
d) P(A|B) + P(¬A|B) = P(B)
3. The Theorem of Total Probability states that P(B) can be computed as:
a) ∑ P(Aᵢ|B)
b) ∑ P(Aᵢ, B) [rationale: P(B) = ∑ P(Aᵢ, B) over all i. Since P(Aᵢ, B) = P(B|Aᵢ)P(Aᵢ), this is
the standard form of the theorem.]
,c) ∏ P(B|Aᵢ)
d) max(P(B|Aᵢ))
4. If two events are independent, what is P(A|B)?
a) 0
b) P(A) [rationale: Independence means the occurrence of B provides no information
about A, so the conditional probability is the same as the prior probability.]
c) P(B)
d) P(A,B)
5. You have a robot that can be in one of three possible locations: {1, 2, 3}. Your
belief is [0.4, 0.4, 0.2]. What is the probability the robot is in location 1 or 2?
a) 0.4
b) 0.6
c) 0.8 [rationale: Since these are mutually exclusive states, the probability of being in
either state is the sum: 0.4 + 0.4 = 0.8.]
d) 1.0
6. A bag contains 5 red balls and 5 blue balls. You draw one ball, do not replace
it, and then draw another. The probability of drawing a red ball on the second
draw given the first was red is:
a) 5/10
b) 4/9 [rationale: After drawing one red ball, 4 reds and 5 blues remain (9 total). The
conditional probability is therefore 4/9.]
c) 5/9
d) 1/2
7. In the context of robotics, a "belief" is formally defined as:
a) The exact state of the robot
b) A probability distribution over all possible states [rationale: In probabilistic
robotics, the robot rarely knows its exact state. A belief is the posterior probability
density function over the state space conditioned on all available data.]
c) The most likely state of the robot
d) A measure of the robot's confidence in its sensors
, 8. What is the probability of the intersection of two independent events with
probabilities 0.3 and 0.5?
a) 0.8
b) 0.2
c) 0.15 [rationale: For independent events, P(A,B) = P(A) * P(B) = 0.3 * 0.5 = 0.15.]
d) 0.02
9. A test for a rare disease is 99% accurate. If 1 in 10,000 people have the
disease and you test positive, Bayes' rule helps us understand that the
probability you actually have the disease is:
a) 99%
b) About 50%
c) Close to 1% [rationale: Due to the low prior probability, even a good test yields a
low posterior probability. This is the classic "false positive" paradox, which Bayes'
Rule solves.]
d) 99.99%
10. If P(A) = 0.6 and P(B|A) = 0.5, what is P(A,B)?
a) 0.5
b) 1.1
c) 0.3 [rationale: By the chain rule, P(A,B) = P(B|A) * P(A) = 0.5 * 0.6 = 0.3.]
d) 0.03
11. In probability notation, what does the symbol ¬ represent?
a) And
b) Or
c) Not [rationale: ¬A is the complement of event A, meaning "not A".]
d) Given
12. For mutually exclusive and exhaustive events Aᵢ, which of the following is
equivalent to P(B)?
a) P(B|Aᵢ) for a random i
b) ∑ P(Aᵢ) * P(B|Aᵢ) [rationale: This is the mathematical expression of the Theorem of
Total Probability, which allows us to compute the probability of an event by summing
TestCourse
Instructor: Based on the work of Sebastian Thrun
Total Questions: 85
Instructions: Choose the best possible answer for each question.
Section 1: Probability Fundamentals (Questions 1-15)
1. In probability, the product rule, P(A,B) = P(A|B)P(B), is also known as:
a) The Theorem of Total Probability
b) Bayes' Rule
c) Chain Rule [rationale: The chain rule describes how to calculate the probability of
the intersection of multiple events by multiplying conditional probabilities.]
d) The Law of Large Numbers
2. Which of the following statements is always true for any two random events
A and B?
a) P(A|B) = P(B|A)
b) P(A,B) = P(A) * P(B)
c) P(A ∪ B) = P(A) + P(B) - P(A,B) [rationale: This is the fundamental principle of
inclusion-exclusion for the union of two events.]
d) P(A|B) + P(¬A|B) = P(B)
3. The Theorem of Total Probability states that P(B) can be computed as:
a) ∑ P(Aᵢ|B)
b) ∑ P(Aᵢ, B) [rationale: P(B) = ∑ P(Aᵢ, B) over all i. Since P(Aᵢ, B) = P(B|Aᵢ)P(Aᵢ), this is
the standard form of the theorem.]
,c) ∏ P(B|Aᵢ)
d) max(P(B|Aᵢ))
4. If two events are independent, what is P(A|B)?
a) 0
b) P(A) [rationale: Independence means the occurrence of B provides no information
about A, so the conditional probability is the same as the prior probability.]
c) P(B)
d) P(A,B)
5. You have a robot that can be in one of three possible locations: {1, 2, 3}. Your
belief is [0.4, 0.4, 0.2]. What is the probability the robot is in location 1 or 2?
a) 0.4
b) 0.6
c) 0.8 [rationale: Since these are mutually exclusive states, the probability of being in
either state is the sum: 0.4 + 0.4 = 0.8.]
d) 1.0
6. A bag contains 5 red balls and 5 blue balls. You draw one ball, do not replace
it, and then draw another. The probability of drawing a red ball on the second
draw given the first was red is:
a) 5/10
b) 4/9 [rationale: After drawing one red ball, 4 reds and 5 blues remain (9 total). The
conditional probability is therefore 4/9.]
c) 5/9
d) 1/2
7. In the context of robotics, a "belief" is formally defined as:
a) The exact state of the robot
b) A probability distribution over all possible states [rationale: In probabilistic
robotics, the robot rarely knows its exact state. A belief is the posterior probability
density function over the state space conditioned on all available data.]
c) The most likely state of the robot
d) A measure of the robot's confidence in its sensors
, 8. What is the probability of the intersection of two independent events with
probabilities 0.3 and 0.5?
a) 0.8
b) 0.2
c) 0.15 [rationale: For independent events, P(A,B) = P(A) * P(B) = 0.3 * 0.5 = 0.15.]
d) 0.02
9. A test for a rare disease is 99% accurate. If 1 in 10,000 people have the
disease and you test positive, Bayes' rule helps us understand that the
probability you actually have the disease is:
a) 99%
b) About 50%
c) Close to 1% [rationale: Due to the low prior probability, even a good test yields a
low posterior probability. This is the classic "false positive" paradox, which Bayes'
Rule solves.]
d) 99.99%
10. If P(A) = 0.6 and P(B|A) = 0.5, what is P(A,B)?
a) 0.5
b) 1.1
c) 0.3 [rationale: By the chain rule, P(A,B) = P(B|A) * P(A) = 0.5 * 0.6 = 0.3.]
d) 0.03
11. In probability notation, what does the symbol ¬ represent?
a) And
b) Or
c) Not [rationale: ¬A is the complement of event A, meaning "not A".]
d) Given
12. For mutually exclusive and exhaustive events Aᵢ, which of the following is
equivalent to P(B)?
a) P(B|Aᵢ) for a random i
b) ∑ P(Aᵢ) * P(B|Aᵢ) [rationale: This is the mathematical expression of the Theorem of
Total Probability, which allows us to compute the probability of an event by summing
