PHIL 7001 / Fall 2024 Professor Boris Babic
PHIL 7001: Homework Assignment 1
Professor Boris Babic
Due date: September 25th, 2025, 11:59pm
Instructions:
This is the first homework assignment for PHIL 7001. The assignment will be graded
out of a total of 100 points. It is open notes, and open book – this means that you can
use any resources you like in order to formulate your answer. However, please do not
“copy” from internet materials.
Course materials: You can directly use course materials – both lectures and readings –
without citation.
Non course materials: If you directly rely on external sources from the internet or other
textbooks or articles, then please provide a citation to those references.
In order to write your answers you can use a word processor, or (for mathematics) you
can write your answers in pen and then scan them into a pdf. If you know how to
use LaTeX, then you can also use LaTeX. But if you don’t know how to use it, then
you don’t have to. Whichever method you choose, please ensure that you submit your
homework assignment via the course moodle page as one single PDF. Please make
sure that everything within this single PDF is clearly legible. This will make it much
easier for grading purposes.
Calculators are allowed, and you can use any calculator you wish.
Please write your name and student number clearly on the first page of your
submission, and include page numbers at the bottom of each page of your submis-
sion.
There should be no trick questions on the homework. If anything seems unclear or
ambiguous, then you can use your judgment to resolve the ambiguity, and note that
you have done so in your answer. For example: if you are not sure whether to write
something mathematically, or in words, it would be safe to do both, or to pick one
option and explain why you picked it (of course: it is always best to just ask me for
clarification! But if you cannot ask me, and you need to answer, then this is what I
recommend).
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PHIL 7001 / Fall 2024 Professor Boris Babic
Problem 1: Probability Basics (20 points)
a. State or describe the (three) conditions that a function, P (A), must fulfill in order
to be a legitimate probability function.
b. Suppose we have three events (A, B, C) and we have the following vector of
illegitimate probability assignments for A, B, and C, respectively: (.6, .8, .6).
Why is this vector of probability assignments illegitimate? What would you do
to it in order to make it a legitimate assignment of probabilities?
c. Using the formula for the probability of a union, calculate P (A∪B) when P (A) =
0.3, P (B) = 0.5, and P (A ∩B) = 0.1.
d. If P (A) = 0.6, P (B) = 0.8, and P (A∩B) = 0.5, what is the probability of neither
event A nor event B occurring?
e. Calculate the probability of event A occurring given that event B has occurred,
if P (A) = 0.4, P (B) = 0.3, and P (A ∩B) = 0.15.
For part (e), it will help you to know that P (A|B) = P (A∩B)
P (B)
.
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PHIL 7001 / Fall 2024 Professor Boris Babic
Problem 2: Bayes’ Rule (20 points)
Suppose that the influenza virus (i.e., the flu) this winter in Hong Kong has an occur-
rence rate of 0.01 (i.e., 1%). Suppose that a very reliable test has been developed to
check whether or not someone has the flu.
The test has a sensitivity rate of 0.96 (i.e., 96%). This means that:
Pr( Positive Test | Patient has Disease) = 0.96.
The test has a specificity rate of 0.97 (i.e., 97%). This means that:
Pr(Negative Test | Patient does not have Disease) = 0.97.
Find the probability that a patient has the disease, given that they receive a positive
test. That is, find:
Pr(Patient has Disease | Positive Test).
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PHIL 7001 / Fall 2024 Professor Boris Babic
Problem 3: Probability Spaces (30 Points)
a. Write down the binomial distribution in mathematical form and for each term in
the distribution, state whether it is a random variable, or a parameter.
b. Suppose it is known that 90% of students at HKU prefer white sneakers to black
sneakers. If you ask 6 randomly selected students on campus, what is the proba-
bility that exactly 3 of them prefer white sneakers to black sneakers? If you ask
10 randomly selected students on campus, what is the probability that exactly 3
of them prefer white sneakers to black sneakers?
c. Suppose that 30% of people in this class are night owls (prefer to sleep late, wake
up late). If you ask 10 random people in the class if they are a morning person
or night owl, what is the probability that more than 8 people will say they are a
morning person?
d. Write down the normal distribution in mathematical form, and for each term in
the distribution, state whether it is a random variable, a parameter, or a constant.
e. Suppose we have a normally distributed random variable, X, whose mean is 17
and standard deviation is 3. You want to find the probability that this random
variable takes on a value between 14 and 20.
How would you calculate this using the python function norm.cdf(x, loc, scale),
as discussed in class? (in other words, write down the correct expression using
the norm.cdf function)
f. Suppose that a certain population follows a normal distribution with mean 10
and standard deviation of 2. State the approximate probability that a randomly
selected value is greater than 14, and explain your answer.
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PHIL 7001 / Fall 2024 Professor Boris Babic
Problem 4: AI applications of Bayes’ Rule (30 Points)
You are analyzing the fairness of a recidivism assessment algorithm applied to two
demographic groups. The algorithm has been constructed to maintain a precision of
0.8 for both groups. Each group has 1,000 individuals, with the following base rate
of recidivism from historical data:
• Group A: Base rate of reoffending = 0.7 (70% actually reoffend)
• Group B: Base rate of reoffending = 0.2 (20% actually reoffend)
The algorithm produces the following True Positive counts:
• True Positives in Group A (TPA) = 640 people
• True Positives in Group B (TPB) = 100 people
a. Using the precision formula, derive the general relationship between False Posi-
tives (FP) and True Positives (TP) needed to achieve a precision of 0.8.
b. Calculate the required False Positives for each group:
• FPA =
• FPB =
c. Complete the confusion matrix for Group A:
Predicted
Actual High Risk Low Risk Total
Reoffend
No Reoffend
Total 1000
1. c. Complete the confusion matrix for Group B:
Predicted
Actual High Risk Low Risk Total
Reoffend
No Reoffend
Total 1000
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