Instructor: James Schmidt
Email: james.a.schmidt.gr@dartmouth.edu
Time: 12:50-1:55 MWF (room 105)
X-hour: 1:20-2:10 Tu  (room 105)

Peer Tutoring: 7-8 M, 8-9 W, 2-3 Sun (Dartmouth Hall 002)

TA Helproom: 7-9 Tu/Th/Sat (room 105)
Office hours: After Class

Textbooks: 

Introduction to ProbabilityLinks to an external site., by Joseph Blitzstein and Jessica Hwang  

Introduction to Probability, by Grinstead and Snell (answers here) 

Probability is the study of random events and the likelihood that they occur. A fair coin comes up heads half the time, but how likely are you to see exactly 50 heads in 100 tosses? Why is a full house worth more than a flush in poker? What exactly is a ‘bell curve’, and why do we care so much about it? We will discuss these questions and more.

Selected topics: Probability axioms, counting, random variables, discrete and continuous distributions, expectation, moments, joint distributions, independence, generating functions, limit theorems.

Here is a rough weekly syllabus:

1. Basics (B&H 1.1, 1.2, 1.3, 1.4, 1.5)  (G&S 1.1, 1.2)

2. Counting, inclusion exclusion, probability axioms (B&H 1.6, 2.1, 2.2, 3.1) (G&S 3.1, 3.2, 3.3, 4.1)

3. Conditional probability, Bayes’ Theorem, Independence

4. (MIDTERM 1) Distributions -- Binomial, geometric, etc 

5. Negative binomial, Poisson, expectation, variance

6. Continuous random variables

7. (MIDTERM 2) More continuous random variables

8. Moments

9. Joint distributions, inequalities and limit theorems

10. whatever tickles my fancy

This can change, but hopefully not too much.

We will use X-hours sporadically for class. If the X-hour is not being used for class, then it will be used as office hours.

Grade Breakdown: 

10% homework

30% for each exam

Homework:

Homework will be assigned at the end of every class and will be due at the beginning of every class. It can be handed in either digitally via email or physically on paper. Homework will be graded a 0 for absent or totally wrong, 1 for on the right track, and 2 for correct. Each day's homework will also be listed here. 

• Monday, June 29: What is/are your major/majors? And what times work for you for office hours this term?
• Tuesday, June 30: You roll three six-sided dice, one at a time. Each time a roll is less than the previous one, you reroll it until it's at least as much as the previous roll. What is the probability that the three values sum to at least 12? 
  
  
• Wednesday, July 1: You and a friend are playing a game of war with a normal deck of 52 cards. Each round, the first player draws the top card and the second player draws the next card, and then they compare values. The higher value (aces high) wins. (Ties don’t matter.) You see two of the first three cards — they’re a two and a king — though the first three cards are then shuffled so while the top three do contain a two and a king, the third card is unknown and they are in a random order. You play to two points, meaning you win if you win two rounds. If, for whatever reason, you run out of cards, you just reshuffle the whole deck and continue. What is the probability of winning if you go first? What about if you go second? And should you go first or second? 
  • Solution, as per popular demand: As each round, excluding the second (which must always be played) is "fair", meaning that the probability of winning is the same as the probability of losing, we may easy condition on the result of that second round, playing the rest of the game thereafter, as each successive point has a 50/50 chance of being awarded to either player. As such, tying the second round gives a 50% chance of winning, winning the second round gives a 75% chance of winning, and losing the second round gives a 25% chance of winning. If you go first, then you have a 33% chance of getting a king (where you win the round with probability 43/50, tie with 3/50, and lose with 4/50), a 33% chance of getting a two (where you win with probability 0/50, tie with 3/50, and lose with 47/50), and a 33% chance of having a fair round (in which case the total odds of winning are 50%). Summing this all up gives the chance of winning as 73/150. Going second, therefore, gives the chance of winning as 77/150, which is better by a slim margin.
• Monday, July 6: Finish the calculation at the end of class to determine the probability of n drunken Dartmouth students each walking off without their own jackets, aka a "derangement". Recall that we used the Principle of Inclusion and Exclusion to determine that it was the following: 
  • $\sum_{k=1}^n(-1)^{k-1}\binom{n}{k}\left(1-\frac{(n-k)!}{n!}\right)$

If you collaborate with anyone in the class, you each must submit individual assignments in your own words, and must list all collaborators. 

Exams: 

All exams will require you to check with your phone and student ID. 

Please be aware that there will be no note sheets or electronic devices allowed on your person while taking an exam. 

Midterms will take place on the Wednesdays of their respective weeks, proceeded by an X-hour review session. 

Generative AI:

AI can likely solve every problem you're going to see in this course. As you will likely want to pass this course, it is thus highly beneficial if you do not use AI to solve the problems here. 

Accessibility Services

If you need any accommodations for the course, please get in touch with me and Student Accessibility Services as soon as you can! In particular, we will need to figure out accommodations for our quiz structure.

Academic Honor Principle

You should be advised of Dartmouth’s Academic Honor Principle. For this course, the important rules are:

• No outside help on exams or quizzes. You should only have something to write with, the paper, and your brain.

• Only turn in homework which reflects your own understanding of the material. Feel free to work with other people, consult the internet, books, and so on, but whatever you turn in must be your own independent work.

 

Much of the formatting and text here stolen shamelessly from Robert Dougherty-Bliss