2025 December 2,

Math 240: Probability

Carleton College, Fall 2025, Dr. Joshua R. Davis, , CMC 324, x4095

Introduction

Probability is a beautiful subject of pure mathematics. It provides the conceptual foundation for statistics and machine learning. It has applications to many other disciplines: physics, chemistry, computer science, finance, etc.

This course is an introduction to probability. Approximately half of the content is discrete and the other half continuous. We do some simulations and other exercises using the free software R. The prerequisites are Math 120 or Math 211. Talk to me if you are concerned about your background.

Responsibilities

The College's accreditation says that a 6-credit course is 150 hours of work. That's about 15 hours per week or 5 hours per class meeting. Those 5 hours break down into about 1 hour for class itself and 4 hours for homework, reading, studying, etc. If you find yourself spending much more time, then talk to me.

Participation

Our course meets in CMC 301 during period 2A. You are expected to attend every class meeting promptly. You are expected to take notes on paper or a tablet (not a laptop). You are expected to participate in discussion and group work.

Phones and laptops are not permitted. Photographs, audio recordings, and video recordings are not permitted. Why? Because we all work together to maintain a distraction-free environment. Because taking notes is much more educational than taking photographs. Because typing on a laptop can't easily capture the math notation and drawings that are needed.

During the first two weeks of the course, you are required to visit me in my office. Why? Because otherwise I have students who never speak to me. That's bad for education.

Some students don't like participating in class, because they are shy, they are not confident about their English or math, etc. I urge those students to participate anyway. They can also compensate for a deficiency in class participation by collaborating with others in office hours.

Homework

Although class meetings may seem like the core of the course, homework assignments are actually where you learn the material. It is essential that you attempt each homework promptly, before the next class meeting. For then you better understand that next class meeting!

On homework, you are encouraged to figure out the problems with other students. However, you should always write/type your solutions individually, in your own words. You may not copy someone else's work — that includes artificial intelligence products — or allow them to copy yours. Presenting someone else's work as your own is an act of academic dishonesty. The College requires me to report you, if I suspect that you have not upheld its Academic Integrity standards.

Writing is not just for literature and history majors. Written and oral communication skills are essential to every academic discipline and are highly prized by employers. In this course, your written work is evaluated both for correctness and for presentation. Compose your solutions as if the intended audience is your fellow students. By doing so, you show enough detail that the grader can ascertain whether you yourself understand the material.

Homework is assigned at nearly every class meeting. Although you are expected to attempt the problems immediately, they are usually collected two meetings after they were assigned. This schedule tries to give you some flexibility.

But we all have bad weeks, where we can't get everything done, right? If you need to submit an assignment late, then do so. Put your paper in a separate pile from the on-time papers. If the grader hasn't graded the assignment yet, then they can grade your paper with the others for full credit. If the grader has graded the assignment already, then you might not get credit. There are limitations to how much delay and complication a grader can handle.

Please mark each assignment with the day that it was assigned (e.g., "Day 11"). Please staple multi-page packets. Paper clips don't work well in a stack of papers.

Depending on time constraints in any given week, perhaps not all of your homework will be graded. In order to ensure full credit, do all of the assigned problems by their due date.

Exams

We have four exams. The first three exams are given in class, and the fourth exam is given during our official final exam slot. See the schedule below.

Are the exams cumulative? No and yes. No, in that most exams are focused on the material that has not been tested yet. Yes, in that much of the material is inherently cumulative. Also, the last exam will have some questions that explicitly address material from earlier in the course.

Grading

For better or worse, we are required to measure your learning using grades. Your numerical grade is based on the responsibilities above: participation 5%, homework 10%, Exam A 15%, Exam B 20%, Exam C 20%, Exam D 30%.

Numerical grades are converted to letter grades only at the end of the term. Grades are not curved, so students are not in competition with each other. Grades are also not based on predetermined percentages (90%, 80%, 70%, etc.), because Math 240 problems are difficult to tune so precisely. (I could accidentally write a difficult exam and wreck everyone's percentages.) Rather, I assign grades by comparing the students to the course goals. For example:

• A student who earns an A isn't perfect, but they can complete almost all problems and speak fluently about nearly all of the content.
• A student who earns a B learns most of the big concepts pretty well, but they are missing some important problem-solving techniques.
• A student who earns a C gets some of the material but has significant gaps in their understanding of major concepts or their ability to solve typical problems.

Talk to me, if you are concerned about your grade. :)

Resources

I want all of my students to work hard and learn a lot. I try to give them all of the resources that they need. For starters, the course materials are

• Probability with Applications and R by Wagaman and Dobrow, 2nd edition. This text is used in many, but not all, sections of Math 240 at Carleton.
• Classwork lists many of the problems that we study in class.
• Poisson Processes summarizes Sections 3.5, 6.5, and 7.3 in a coherent manner.
• R Instructions help you run R/RStudio either in a web browser or on your own computer.
• Calculus Review summarizes derivatives, anti-derivatives, single and double integrals, and series.
• Challenge Problems are some unusual exercises, for students who are feeling ambitious.
• Here are the exams from this term, with quartiles (75th, 50th, and 25th percentiles) of student scores.
  • Exam A: Problems, Solutions (56/60, 50/60, 40/60)
  • Exam B: Problems, Solutions (36/52, 32/52, 26/52)
  • Exam C: Problems, Solutions (34/43, 30/43, 25/43)
  • Exam D: Problems, Solutions (58/63, 51/63, 48/63)
• Here are the exams from my winter 2021 version of this course. Keep in mind that the order of topics varies somewhat from term to term. Also, these exams were take-home because of the COVID-19 pandemic, whereas our exams are given in class.
  • Exam A Problems, Solutions
  • Exam B Problems, Solutions
  • Exam C Practice, Problems, Solutions
• Here are the exams from my fall 2019 version of this course. They were all given in class.
  • Exam A: Exam, Solutions (40/49, 36/49, 34/49)
  • Exam B: Exam, Solutions (42/48, 39/48, 34/48)
  • Exam C: Exam, Solutions (52/61, 49/61, 42/61)

Remember that I encourage you to solve problems with classmates (even if the work that you submit must be your own). If you want help in finding a study partner, then e-mail me, perhaps describing some of your habits: working in the middle of the night, not waiting until deadlines, etc.

My office hours are Mon 1:50-3:00 (5A), Tue 10:10-11:00, Wed 11:10-12:20 (3A), Thu 9:30-10:20. Office hours are essentially optional extra class meetings, where you pick the topic of conversation — usually homework problems. No appointment is needed for office hours; just drop in! If you want to visit office hours but the times don't work, then consult my weekly schedule and e-mail me, listing several possible meeting times.

You can also get drop-in help from more experienced students. It's available Sunday-Friday, 7-10 PM, in CMC 304. They can help you with probability and with R.

If a health condition or other personal matter affects your participation in class, homework, exams, etc., then please let me know as soon as possible. Depending on the situation, we might want to confer with Accessibility Resources, Assistive Technology, Student Health and Counseling, or Title IX. When you ask me to help, I do my best to help. :)

Schedule

To help you decode the schedule, here is an example. Monday September 15 is Day 01 of the 28-day course. During class, we discuss basic concepts. You are expected to skim Sections 1.1-1.5 of the textbook before or after class. You have a homework assignment called Day 01. Attempt it immediately, so that you are ready for class on Day 02. Submit the homework for grading at the start of class on Day 03, which is Friday.