University of Washington
Department of Mathematics Homework 3

MATH 394: Probability I

Instructor: Arman Jahangiri

Homework Policy

Submission and Deadline

Please:

• Submit homework as a single PDF on Gradescope by 10:00 PM (Pacific Time) on the listed due date.
• Begin each problem on a new page, and ensure that pages are correctly tagged in Gradescope.
• Write solutions clearly and legibly.
• Typesetting solutions in LaTeX is encouraged, but not required.

Late Days

Each student is allotted six late days for the quarter. A late day extends a homework deadline by up to 24 hours without penalty. For example, submitting an assignment anytime between 10:01 PM on the due date and 10:00 PM the following day counts as one late day.

The following rules apply:

• No explanation is required to use late days.
• At most two late days may be used on any single homework assignment.
• Late days may not be used on the final homework assignment or on any assignment for which solutions have already been released.
• It is the student’s responsibility to keep track of how many late days have been used during the quarter.

Once all late days have been exhausted, additional late submissions will incur a penalty of 10% per day, up to a maximum deduction of 50%. Assignments submitted more than five days late, or after solutions have been released, will not be accepted.

Submission Issues and Technical Difficulties

If a serious technical issue prevents a timely Gradescope submission, students may temporarily submit their assignment by email to the instructor at armanjg@uw.edu. In such cases:

• The submission must consist of a single PDF file. The email subject line must follow the format: \[ \texttt {HW\# Submission - FULL NAME - STUDENT ID}. \]
• Once the Gradescope issue has been resolved, the student must upload the exact same file to Gradescope as soon as possible.
• The student should clearly indicate in the Gradescope submission that the assignment was previously submitted by email before the deadline.

Email submissions are intended only for genuine technical emergencies and should not be used as a substitute for timely Gradescope submission.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 2, Problem 1.

A spam filter is designed by looking at commonly occurring phrases in spam. Suppose that \(80\%\) of email is spam. In \(10\%\) of the spam emails, the phrase “free money” is used, whereas this phrase is only used in \(1\%\) of non-spam emails. A new email has just arrived, which does mention “free money.” What is the probability that it is spam?

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 2, Problem 6.

A hat contains 100 coins, where 99 are fair but one is double-headed, always landing Heads. A coin is chosen uniformly at random. The chosen coin is flipped 7 times, and it lands Heads all 7 times.

Given this information, what is the probability that the chosen coin is double-headed?

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 2, Problem 10.

Fred is working on a major project. In planning the project, two milestones are set up, with dates by which they should be accomplished. This serves as a way to track Fred’s progress. Let \(A_1\) be the event that Fred completes the first milestone on time, \(A_2\) be the event that he completes the second milestone on time, and \(A_3\) be the event that he completes the project on time.

Suppose that \[ P(A_{j+1}\mid A_j)=0.8 \qquad \text {but}\qquad P(A_{j+1}\mid A_j^c)=0.3 \] for \(j=1,2\), since if Fred falls behind on his schedule it will be hard for him to get caught up. Also, assume that the second milestone supersedes the first, in the sense that once we know whether he is on time in completing the second milestone, it no longer matters what happened with the first milestone. We can express this by saying that \(A_1\) and \(A_3\) are conditionally independent given \(A_2\), and they are also conditionally independent given \(A_2^c\).

(a)

Find the probability that Fred will finish the project on time, given that he completes the first milestone on time. Also find the probability that Fred will finish the project on time, given that he is late for the first milestone.

(b)

Suppose that \(P(A_1)=0.75\). Find the probability that Fred will finish the project on time.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 2, Problem 20.

The Jack of Spades, Jack of Hearts, Queen of Spades, and Queen of Hearts are taken from a deck of cards. These four cards are shuffled, and then two are dealt. Literary references to cider, tarts, and winks do not need to be considered when solving this problem.

(a)

Find the probability that both of these two cards are queens, given that the first card dealt is a queen.

(b)

Find the probability that both are queens, given that at least one is a queen.

(c)

Find the probability that both are queens, given that one is the Queen of Hearts.

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 2, Problem 30.

A family has 3 children, creatively named \(A\), \(B\), and \(C\).

(a)

Discuss intuitively, but clearly, whether the event “\(A\) is older than \(B\)” is independent of the event “\(A\) is older than \(C\).”

(b)

Find the probability that \(A\) is older than \(B\), given that \(A\) is older than \(C\).

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 2, Problem 31.

Is it possible that an event is independent of itself? If so, when is this the case?

Source: Blitzstein and Hwang, Introduction to Probability, Chapter 2, Problem 35.

You are going to play 2 games of chess with an opponent whom you have never played against before. Your opponent is equally likely to be a beginner, intermediate, or master. Depending on which, your chances of winning an individual game are \(90\%\), \(50\%\), or \(30\%\), respectively.

(a)

What is your probability of winning the first game?

(b)

Congratulations: you won the first game! Given this information, what is the probability that you will also win the second game, assuming that, given the skill level of your opponent, the outcomes of the games are independent?

(c)

Explain the distinction between assuming that the outcomes of the games are independent and assuming that they are conditionally independent given the opponent’s skill level. Which of these assumptions seems more reasonable, and why?