Homework #1
EE 364: Spring 2026
Assigned: 15 January
Due: Thursday, 22 January at 16:00
BrightSpace Assignment: Homework 1
Write your solutions to these homework problems. Submit your work to BrightSpace by the due date. Show all work and box answers where appropriate. Do not guess.
Problem 0
Daily derivation #1, \(P(A) + P(B) = P(A \cup B) + P(A \cap B)\).
Problem 1
Prove or disprove the following statements.
\([A \subset B] \Longrightarrow [A \cap B = A]\).
\([A^c \cup B = \Omega] \Longrightarrow [A \subset B]\).
\([A \subset B] \Longrightarrow [A^c \cup B = \Omega]\).
From (b) and (c) above can you justify that \([A \subset B]\) and \([A^c \cup B = \Omega]\) are equivalent? Two statements are equivalent if they always have the same truth value (either both true or both false but never different). Explain.
Problem 2
Prove or disprove the following statements. You must prove any set theoretic theorems that you use. That includes set associativity, commutativity, distributivity, De Morgan’s law, etc.
\(A \cap B = \emptyset\) if and only if \(A \cup B = (A \cap B^c) \cup (A^c \cap B)\).
\((A \cap B) - C = (A - C) - (B - C)\).
\((A^c \cup B)^c \cap A^c = \emptyset\).
Problem 3
Suppose \(\Omega = \{w, x, y, z\}\).
How many distinct subsets does \(\Omega\) have? List them.
If \(A = \{w, x\}\) how many supersets does \(A\) have in \(\Omega\)? List them.
If \(A = \{u, w, y\}\) how many supersets does \(A\) have in \(\Omega\)? List them.
Problem 4
The sample space describes the set of all possible outcomes of a random experiment. The cardinality of \(\left| \Omega \right|\) is the number of elements in \(\Omega\). The power set \(2^{\Omega} = \{A: A \subset \Omega\}\) is a collection of all the subsets of \(\Omega\).
How big is the power set (i.e. cardinality, \(\left| 2^\Omega \right|\)) of each of the following sets?
\(\Omega = \{H,T\}\).
The sample space represented by the outcome of a single roll of a six-sided die.
The set of positive integers less than or equal to \(N\), \(\{1, 2, \ldots, N\}\).
Write the power set for each of the following sample spaces:
\(\Omega = \{a\}\).
\(\Omega = \{a, b\}\).
\(\Omega = \{a, b, c\}\).
Prove that if \(A \subset B\) then \(2^A \subset 2^B\).
Problem 5
Mary chooses a letter of the alphabet (a-z) at random. Describe the sample space \(\Omega\) and probability measure \(P\). Compute the probability that a vowel (a, e, i, o, u) is sampled.
A collection of plastic letters (a-z) is mixed in a jar. John draws two letters at random and without replacement (one after the other). Describe the sample space \(\Omega\). What is the probability that John draws a vowel (a, e, i, o, u) and a consonant? What is the probability that John draws two vowels?
Problem 6
Prove: If \(A \subset B\) then \(P(B - A) = P(B) - P(A)\).
Prove: \(P(A) = P(B)\) if and only if \(P(A \cap B^c) = P(A^c \cap B)\).
Problem 7
Events \(A\) and \(B\) have probabilities \(P(A) = \frac34\) and \(P(B) = \frac13\).
Show that \(\frac{1}{12} \le P(A \cap B) \le \frac{1}{3}\).
Describe sample spaces and events \(A\) and \(B\) having \(P(A) = \frac34\) and \(P(B) = \frac13\) such that:
\(P(A \cap B) = \frac{1}{12}\).
\(P(A \cap B) = \frac{1}{3}\).
Problem 8
Events \(A\) and \(B\) satisfy \(P(A \cup B) = 0.8\) and \(P(A \cap B) = 0.3\).
What are the tight bounds on \(P(A)\)?
If additionally \(P(A) = 0.6\), compute \(P(B)\), \(P(A^c \cap B)\), and \(P(A \cup B^c)\).
Events \(A\), \(B\), \(C\) satisfy \(P(A) = P(B) = P(C) = 0.5\) and \(P(A \cap B) = P(A \cap C) = P(B \cap C) = 0.2\).
Find tight bounds on \(P(A \cap B \cap C)\).
Find tight bounds on \(P(A \cup B \cup C)\).
Construct sample spaces achieving each extreme.
Problem 9
An urn contains \(r\) red balls and \(b\) blue balls. Two balls are chosen at random without replacement.
Let \(R\) be the event “both balls are red.” Express \(P(R)\) in terms of \(r\) and \(b\).
Let \(S\) be the event “both balls are the same color.” Express \(P(S)\) in terms of \(r\) and \(b\).
For what ratio \(r/b\) is \(P(S)\) minimized?