## Probability Theory (AMS 311) Course Material, Fall 2013

### Joe's Office Hours (tentative):

Tues (10:00-11:15) and Wed (2:00-3:30), and please drop by any time, or send email or call

### Teaching Assistants:

All TA office hours are in Harriman 010 (AMS Help Room).
Xiaohan Zhang (xiaohan.zhang@stonybrook.edu), Office hours: Mon/Wed, 1:30-2:30;
Jingwen Che (jingwen.che@stonybrook.edu), Office hous: Mon 2:30-3:30; Wed 3:00-4:00

## Announcements:

Reminder; Exam2 is in class on Thursday, Nov 7. Review will take place on Tues, Nov 5 -- be sure to be there!

### Main Course Information

Course Information (AMS 311), Fall 2013 This is the main course information sheet with details about exams, homeworks, grading, etc.

### Homeworks and Other Handouts

Homework 1, due Thursday 9/12; Solutions are posted on Blackboard after the due date.
Examples on conditional probability
Homework 2, due Thursday 9/19; Solutions are posted on Blackboard after the due date.
Notes and examples on discrete distributions
Homework 3, due Thursday 9/26; Solutions are posted on Blackboard after the due date.
Notes on continuous distributions
Homework 4, due Thursday, 10/17. Solutions are posted on Blackboard after the due date.
Homework 5, due Thursday, 10/24. Solutions are posted on Blackboard after the due date.
Examples of joint distribution problems, some solved in class. You should be able to solve all of them. Solution Notes.
Homework 6, due Thursday, 10/31. Solutions are posted on Blackboard after the due date.
Notes on Expectation, Moment Generating Functions, Variance, and Covariance
Cover Sheet for Exam2, showing the formulas and instructions.
Computing Expectations and Probabilities by Conditioning
Homework 7, due Tuesday, 11/19. Solutions are posted on Blackboard after the due date.
Homework 8, due Tuesday, 11/26. Solutions are posted on Blackboard after the due date.

Notes on the Chebyshev Inequality
Homework 9, due Thursday, 12/5. Solutions are posted on Blackboard after the due date.

### An On-Line Interactive Textbook with Examples

Virtual Laboratories in Probability and Statistics, by Kyle Siegrist. Very nice and extensive interactive "text" on probability and statistics. Lots of examples.

### Links of Interest to the Course:

The Infinite Actuary, a site that gives lessons and sample exams for the actuarial exams (including the "P" exam that covers many topics from this course)

Compendium of Probability Distributions
Markov chain applet, to build and simulate chains (can do rat in maze, etc)
Let's Make a Deal applet

Braingle, probability brain teasers
Brief report on Monty Hall problem, by Afra Zomorodian
Probability by Surprise
Cut-the-Knot Probability Page
Chance Web Page, real-life uses of probability and statistics (co-authored by my office-mate from Stanford, Bill Peterson)
Nick's Mathematical Puzzles, with several nice probability puzzles, including some classics!
Conditional Probability Discussion

Let's say you have 5 cents (call it "5c", for short). Let's turn it into 50 cents! First, we know that 5c = sqrt(25 c), right? ("sqrt" is square root) And 25c is the same as $(1/4), a quarter dollar. Now, sqrt($(1/4)) = $(1/2), a half dollar, of course. And$(1/2) is the same as 50 cents. So, in total, we have proven that 5 cents is the same as 50 cents -- a great way to make money! In summary: 5c = sqrt(25c) = sqrt($(1/4)) =$(1/2) = 50c.

I know two distinct secret numbers: call them X and Y, and assume that X < Y, without loss of generality. You have no clue how I came up with them. They could be anything (positive, negative, rational, irrational, etc). They could come from any probability distribution (discrete or continuous). You have no idea. I flip a fair coin. If the coin shows Heads, I reveal to you the larger number, Y; if it shows Tails, I reveal to you the smaller number, X. You do not get to see the result of the coin flip. Your goal is to guess whether the coin was Heads or Tails, based only on your seeing the one number that I revealed to you. Obviously, if you just decide Heads'' is your guess, without taking into account the revealed number at all, then you are correct with probability 0.5. But your goal is to be able to be correct with probability {\em strictly greater} than 0.5. Devise a method to do this, and explain your solution.

The three way pistol duel puzzle: You're a cowboy, and get involved in a three way pistol duel with two other cowboys. You are a poor shot, with an accuracy of only 33%. The other two cowboys shoot with accuracies of 50% and 100%, respectively. The rules of the duel are one shot per cowboy per round. The shooting order is from worst shooter to best shooter, so you get to shoot first, the 50% guy goes second, and the 100% guy goes third, then repeat. If a cowboy is shot he's out for good, and his turn is skipped. Where or who should you shoot first?

You have two ropes. Each takes exactly 60 minutes to burn. They are made of different material so even though they take the same amount of time to burn, they burn at separate rates. In addition, each rope burns inconsistently. How do you measure out exactly 45 minutes? (You are to assume that, while the speed of the rope burning may vary with position x along the rope, the speed of burning to the right at x is the same as the speed of burning to the left at x.)

Two engineering students had thoroughly impressed their Professor in one of their core subjects, having achieved the highest marks throughout the semester's continuous assessments. Being one to give credit where due, the Prof made no secret of his two prize students, continually praising their achievements. The weekend before their final exam (which was to contribute fifty percent of their final grade,) these two geniuses felt they deserved a hard earned break and so went out of town to celebrate their 'pending' success. Having partied late all weekend (and then some,) the two students overslept Monday morning, missing their final exam! Never lacking in will power, they were determined not to fail, having come so far, so well. Upon rolling into town, they appealed to the Prof, and conveniently explained that they had a flat tire while out visiting and that it was not their fault for missing the exam. The Professor was very understanding and rescheduled their exam to the next day, much to the student's delight! They arrived at his office the next morning, where he sat them on each side of his desk facing opposite walls, while he worked on grading the other exams. They opened their examination books and were both pleased with the first easy 10-point question that came directly from a homework assignment. You could hear nothing but the quick writing of their pencils in the room, until they both flipped the page, almost at the same time, where the second question read thusly... "90 points: Which tire?"

A riddle (to illustrate the importance of using UNITS):
How to make 5 cents into 50 cents:
5c = sqrt(25c) = sqrt((1/4)$) = (1/2)$ = 50c