Homework #2

EE 364: Spring 2026

ImportantAssignment Details

Assigned: 22 January
Due: Thursday, 29 January at 16:00

BrightSpace Assignment: Homework 2

WarningInstructions

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 #2, Prove the Theorem of Total Probability and Bayes Theorem for evidence \(E\) and a partition of hypotheses \(H_1, \ldots, H_n\).

Problem 1

Use truth tables to prove whether these propositional assertions are valid or invalid:

  1. \((P \rightarrow \ \sim Q) \Leftrightarrow [(P \vee Q) \wedge \sim (P \vee Q)]\).

  2. \(\sim P \Rightarrow (P \Rightarrow Q)\).

  3. \([P \wedge (Q \rightarrow R)] \Leftrightarrow [(\sim P \vee Q) \rightarrow (P \wedge R)]\).

  4. \([(P \rightarrow Q) \wedge (P \rightarrow \ \sim Q)] \Leftrightarrow \ \sim P\).

Problem 2

The sample space describes the set of all possible outcomes of a random experiment. A sigma algebra is a collection of subsets of the sample space, (i.e. a “set of sets”) that is CUT:

Suppose \(\Omega\) is a sample space. Then \(\mathcal{A}\) is a sigma algebra if \(A \subset \Omega\) for all \(A \in \mathcal{A}\) and

  • C: if \(A \in \mathcal{A}\) then \(A^c \in \mathcal{A}\).
  • U: \(A_1, A_2, \ldots \in \mathcal{A}\) implies \(\bigcup_k A_k \in \mathcal{A}\).
  • T: \(\Omega \in \mathcal{A}\).

  1. Determine if each of these set collections is a sigma algebra.

    1. \(\{\emptyset, \Omega\}\).

    2. \(\{\emptyset, A, A^c, \Omega\}\) where \(A \subset \Omega\).

  2. Suppose \(\Omega = \{a, b, c, d\}\). Is the set collection \(\mathcal{A} = \{\emptyset, \{a\}, \{b\}, \{a, b\}, \{a,c,d\}, \{b,c,d\}, \Omega\}\) a sigma algebra? If not, then could you minimally augment \(\mathcal{A}\) so that it is a sigma algebra.

Problem 3

Use mathematical induction to prove these theorems for all positive integers \(n \ge 1\):

  1. \(11\) divides \(23^n - 1\).

  2. \(\sqrt[n]{n} < 2 - \frac{1}{n}\) for all \(n \ge 2\).

Problem 4

Events \(A\) and \(B\) are independent with \(P(A) > 0\) and \(P(B) > 0\). Which one of the following statements must be true.

  1. \(A - B \ne \emptyset\).

  2. \(A \cap B = \emptyset\).

  3. \(A \cap B \ne \emptyset\).

  4. \(A \cup B = X\).

Problem 5

Consider the six permutations of the letters \(a\), \(b\), and \(c\) as well as the three triples: \(aaa\), \(bbb\), and \(ccc\). Suppose that these nine triples are points in a sample space and assign probability \(1/9\) to each outcome. Let \(A_k\) denote the event that the letter \(a\) is in the \(k\)th position and let \(B_k\) denote the event that the letter \(b\) is in the \(k\)th position.

  1. Are the events \(A_1\) and \(A_2\) independent?

  2. Are the events \(A_1\), \(A_2\), and \(A_3\) independent?

  3. Are the events \(A_1\) and \(B_1\) independent?

  4. Are the events \(A_1\) and \(B_2\) independent?

Problem 6

Suppose \(P(E \cap F) = 0.225\), \(P(F) = 0.5\), and \(P(E|F^C) = 0.1\). Compute \(P(E \cup F)\).

Problem 7

The Los Angeles Metro system conducts a rider survey. Of the 200 riders surveyed, 90 use the Red Line, 70 use the Blue Line, and 60 use the Expo Line. Some riders use multiple lines: 30 use both the Red and Blue Lines, 20 use both the Red and Expo Lines, and 15 use both the Blue and Expo Lines. Only 5 dedicated commuters use all three lines. A Metro official claims that at least 80% of those surveyed use the rail system. Do you agree?

Problem 8

An urn contains 6 red, 4 white, and 2 blue balls. John draws three balls without replacement.

  1. What is the probability that the balls are drawn in the order red, white, blue?

  2. What is the probability that all three balls are the same color?

Problem 9

Mary applies for a competitive internship that requires passing three independent interviews. Based on her preparation, she estimates her probability of passing each interview: 0.7 for the technical screen, 0.8 for the coding challenge, and 0.9 for the behavioral round. She receives an offer if she passes at least one interview.

  1. What is the probability that Mary fails all three interviews?

  2. Let \(A\), \(B\), \(C\) denote the events that Mary passes the technical, coding, and behavioral interviews. Verify that \(P(A \cup B \cup C) = 1 - P(A^c)P(B^c)P(C^c)\).

Problem 10

The events \(A_1\), \(A_2\), and \(A_3\) partition sample space \(X\). \(P(A_1) = 0.3\), \(P(A_2) = 0.2\), and \(P(A_3) = 0.5\). Suppose \(P(E|A_1) = 0.3\), \(P(E|A_2) = 0.5\), and \(P(E|A_3) = 0.8\). What is the probability \(P(A_2|E)\)?

Problem 11

Prove or disprove: If \(P(A) < P(B)\) and \(P(C|A) < P(C|B)\) then \(P(A|C) < P(B|C)\).

Problem 12

John picks up a ripe red apple on the ground and wonders whether he should eat it. The apple has rolled down the hillside from where a clump of three apple trees grow. The three apple trees are genetic clones. But the first tree is twice as old as either of the other two apple trees. The first tree also produces twice as many apples as either of the other two trees produce. Only 10% of the apples on the first (older) tree have worms in them because John sprayed the tree with insecticide. He carefully spared the second tree and so only 5% of its apples have worms. John did not spray the third tree and so half of its apples have worms. What is the probability that the apple that John holds has worms?

Problem 13

A meta-analysis selected 100 studies that measured the relation between the consumption of sugary beverages and obesity. 65 of the studies were sponsored by the food industry and 35 studies had no corporate funding. In the sponsored studies: 15% found an unfavorable result, 22% found a neutral result, and 63% found a favorable result. In the non-sponsored studies: 35% found an unfavorable result, 16% found a neutral result, and 49% found a favorable result. What is the probability that a randomly selected study found a favorable result?

Problem 14

Consider a binary symmetric channel (BSC) with \(P[\textrm{error}] = 0.1\). Suppose that the symbol “0” is sent 4 times as often as symbol “1”. If you receive \(Y = 1\) is it more likely than not that the sent value was “1”?

Problem 15

Machines A, B, and C manufacture bolts in a factory. Machine A produces 25 percent of the total bolts, B produces 35 percent, and C produces 40 percent. Machine A outputs 5 percent defective bolts, B outputs 4 percent defective bolts, and C outputs 2 percent defective bolts. A bolt is drawn at random from the production line. It is defective. What are the probabilities that the defective bolt was manufactured by machines A, B, and C?