Math 30530 - Introduction to Probability

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Homework

Solutions will be posted in a file here, updated week-by-week.

Homework will be announced most Fridays and posted on the course website. It will be due at the beginning of class the following Friday. Each assignment will involve some reading and some problems, possibly on an area not yet covered in lectures. Presented assignments should be neat and legible. At the top of the first page, you should write your name, the course number and the assignment number. If you use more than one page, you should staple all your pages together. The grader reserves the right to leave ungraded any assignment that is disorganized, untidy or incoherent. No late assignments will be accepted. It is permissible (and encouraged) to discuss the assignments with your colleagues; but the writing of each assignment must be done on your own.

Homework 9, due in class Friday April 27:

• Reading: Chapter 3, and Section 4.1.
• Problems to turn in (Numbered problems are taken from the version of the textbook posted on this website):
  1. 3.27
  2. 3.28
  3. 3.33
  4. 4.1
  5. 4.5
  6. 4.6
  7. 4.9 (b) and (c) only
  8. 4.14
  9. 4.18
  10. 4.24
  11. 4.27
  12. Show that if X and Y are independent exponential random variables, both with parameter L, then X+Y is a Gamma random variable with parameters 2 and L.
  13. There are 36 people registered for this class. If everyone was to generate a number uniformly at random from the interval [0,1], what would be the expected value of the smallest number picked?

Homework 8, due in class Friday April 5:

• Reading: Section 2.3.
• Problems to turn in (Numbered problems are taken from the version of the textbook posted on this website):
  1. 2.42
  2. 2.44
  3. 2.46
  4. 2.48
  5. 3.1 (interpretation of part ii: you should no longer assume that (X,Y) is uniform on the four points; what's being asked of you is to find all possible joint distributions that lead to uniform marginals. There may be many of them, and you may need to introduce some variables to describe them all.)
  6. 3.4
  7. 3.5
  8. 3.14
  9. 3.16
  10. 3.19
  11. 3.20
  12. 3.24

Homework 7, due in class Friday March 29:

• Reading: Section 2.3.
• Problems to turn in (Numbered problems are taken from the version of the textbook posted on this website):
  1. 2.31
  2. 2.32
  3. 2.33
  4. 2.34
  5. 2.35
  6. 2.36
  7. 2.37
  8. 2.40
  9. 2.41
  10. 2.43
  11. 2.45
  12. 2.47

Homework 6, due in class Friday March 22:

• Reading: Chapter 2, up to the start of Section 2.3.
• Problems to turn in (Numbered problems are taken from the version of the textbook posted on this website):
  1. 2.6
  2. 2.8
  3. 2.10
  4. 2.11
  5. 2.13
  6. 2.14
  7. 2.16
  8. 2.19
  9. 2.21
  10. 2.26
  11. 2.27
  12. 2.28
  13. 2.30

Homework 5, due in class Wednesday March 6:

• Reading: Chapter 2, up to but not including Poisson pmf.
• Problems to turn in (Numbered problems are taken from the version of the textbook posted on this website):
  1. 2.2
  2. 2.3
  3. 2.4
  4. 2.5
  5. 2.15
  6. 2.17
  7. 2.18
  8. 2.20
  9. 2.22
  10. 2.23
  11. 2.24
  12. 2.25
• Extra credit problem (repeated here in case you didn't pluck it from exam 1). If you are presenting a solution to this problem, you need to justify your answer for any credit!:

  One hundred people line up to board a plane with 100 seats. The first person in line has lost his boarding pass, so he randomly chooses a seat. After that, each person entering the plane either sits in their assigned seat, if it is available, or, if not, chooses an unoccupied seat randomly.

  When the 100th passenger finally enters the plane, what is the probability that she finds her assigned seat unoccupied?

Homework 4, due in class Friday February 22:

• Reading: Chapter 1, through to the end, and Chapter 2, through to page 58.
• Problems to turn in (Numbered problems are taken from the version of the textbook posted on this website):
  1. 1.34 (and also: answer the unconditioned question, ``What is the probability of no 5's'', and then determine whether the event ``no 5's'' is independent of the event ``no 6's'').
  2. 1.36
  3. 1.38
  4. 1.39
  5. 1.42 (Use a formula we proved in class!)
  6. 1.46
  7. 1.48
  8. 1.52
  9. 2.1
  10. 2.2
  11. 2.3
  12. 2.4
• Problems not to turn in (but that might be fun to think about):
  1. An urn has r red and g green balls. Using the law of total probability, it can be shown that if n balls are drawn from the urn and discarded (without their colors being observed), then the probability that the next ball drawn is green is g/(r+b) (the same as it would have been if no balls had initially been drawn). Find a very simple explanation of this, that doesn't need law of total probability.
  2. Here's a game: an ordinary deck of cards is shuffled, and the cards are revealed one after another. At any moment, you get to say STOP, and if the next card turned over is black, you win, while if it is red, you lose. If 51 cards have been turned over, and you still haven't said STOP, you are required to say it at this point (even if you know that the last unturned card is red). Here a strategy that gives you a 50% chance of winning: say STOP before a single card has been turned over. Is there a strategy that gives you a better than 50% chance of winning?

Homework 3, due in class Friday February 15:

• Reading: Chapter 1, through to the end
• Problems to turn in (Numbered problems are taken from the version of the textbook posted on this website):
  1. 1.18
  2. 1.19
  3. 1.20
  4. 1.21
  5. 1.22
  6. 1.24
  7. 1.25
  8. 1.26
  9. 1.27
  10. 1.29
  11. 1.30
  12. 1.33

Homework 2, due in class Friday February 8:

• Reading: Chapter 1, pages 20 through 28
• Problems to turn in:
  1. A dice game is played like this: you roll three dice, and you win if at least one of the following three things happens:
     1. the sum of the three dice is at least 16,
     2. the three values on the dice are all the same, or
     3. all three numbers are even.
     How likely is it that you win the game?
  2. There are 36 people in the class, and I have a jar with 100 quarters.
     1. In how many ways can I distribute quarters to the class? (Students are distinguishable; quarters are not.)
     2. If all ways of distribution are equally likely, how likely is it that everyone gets at least one quarter?
     3. How likely is it that John gets at least one quarter? (John is a particular student in the class.)
     4. How likely is it that everyone gets either 50c or $1?
  3. I toss a fair coin 2n times. I'm interested in the event that exactly half the tosses come up Heads (i.e., that I get n Heads and n Tails). Let p(n) be this probability.
     1. Calculate p(n) for each of n=1,2,3 and 4.
     2. Show that in general, p(n) gets smaller as n gets bigger.
  4. 50 people are called for jury duty. 14 will be on the jury, 8 will be alternates, and 28 will be sent home. All possible selections are equally likely.
     1. How likely are Annie and Bill to both be on the jury?
     2. How likely is it that both are kept (either on the jury, or as alternates, but not necessarily both with same status)?
     3. I know Annie, but I don't know Bill. As I watch people leave the courthouse after selection, I observe that Annie has been sent home. What the probability, given this information, that Bill has been sent home?
  5. 64 players enter a knockout tournament. They are drawn into 32 opening round pairs, in order (so, there's pair no. 1, pair no. 2, et cetera, up to pair no. 32). The first 8 pairs form the first quarter; exactly one of the players in this quarter will get through to the semifinal; the next 8 pairs form the second quarter; the next 8 form the third quarter; and the last 8 form the fourth quarter.
     1. How many opening round draws are possible?
     2. I'm following the fortunes of four players: Caroline, Madison, Angelique and Serena. How likely is it that the four end up in different quarters (and so have a chance to make up the four semifinalists)? (In the end, give an answer in decimal form.) (Hint: try to build a draw in which the four players are in four different quarters, in such a way that it is possible to use the multiplication principle to count the number of such draws.)
  6. Ashley draws two cards, one after another, from a standard deck, with replacement.
     1. How likely is it that both are red?
     2. Ashley gives me the information that at least one of the cards is red. Now, how likely is it that both are red?
     3. Suppose instead that she gives me the information that at least one of the cards is a diamond. Now, how likely is it that both are red?
     4. Suppose instead that she gives me the information that at least one of the cards is the queen of diamonds. Now, how likely is it that both are red?
     5. Is all this surprising?

Homework 1, due in class Friday February 1: (Numbered problems are taken from the version of the textbook posted on this website)

• Reading: Chapter 1 up to page 22.
• Problems to look at, but not to turn in: 1.1, 1.2, 1.3, 1.5, 1.6
• Problems to turn in:
  1. 1.7
  2. 1.8
  3. 1.9 (Hint: the sample space for this experiment consists of all ways to order the four envelopes. Note that you don't know in advance the four amounts in the four envelopes; call those amounts, for example, a, b, c and d, with a < b < c < d.)
  4. 1.10, parts (c) and (d) (The three pieces are placed on distinct squares)
  5. 1.11
  6. 1.12
  7. 1.13, parts (a), (b) and (c)
  8. 1.14
  9. 1.15
  10. 1.16
  11. You can make lots of different pizzas at Blaze, but not all of them are reasonable (a crust with no cheese, no sauce, and just sea salt as a topping, for example, is hardly a reasonable pizza). Say that a pizza is reasonable if it has:
      • One crust from among three different crust options
      • One sauce from among five sauce options
      • Either no cheese, or just vegan cheese, or at most two cheeses from among seven non-vegan cheese options
      • At most three meats from among nine meat options
      • At most three toppings from among fifteen topping options
      • Between meats and toppings, at least one must be chosen
      • Any combination from among three seasoning options
      • Either no finish, or one finish from among seven finishing options.
      1. How many different reasonable pizzas can be made at Blaze?
      2. Based on the information that there are 300 Blaze franchises in existence, the company has been in existence since 2011, Blaze ovens can hold at most eight pizzas at a time, pizzas take three minutes each to cook, and Blaze stores are open at most 12 hours a day, is the following statement plausible: ``There is a reasonable pizza that has never been made at a Blaze store''?

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Exams

The final will be on Tuesday morning May 7, at 8am. Detailed information, including a set of practice questions, and a list of office hours, is available here. Solutions to the practice problems can be found here.

The second midterm exam will be in-class on Monday, April 15. It will cover what we have seen since the first midterm --- essentially, chapters 2 and 3 of the textbook.

• Chapter 2:
  • Discrete random variables & probability mass functions
  • continuous random variables & density functions
  • cumulative distribution functions
  • expectation and variance
  • linearity and monotonicity of expectation
  • Markov's and Chebychev's inequalities
  • law of unconscious statistician
  • probability generating function of discrete random variables
  • the seven basic families of discrete random variables [uniform, Bernoulli, binomial, geometric, hypergeometric, negative binomial, Poisson]
  • the four basic families of continuous random variables [uniform, exponential, normal, gamma]
  • finding the density of a function of a continuous random variable
• Chapter 3:
  • Joint mass function and marginal mass function
  • law of unconscious statistician for joint densities
  • linearity of expectation
  • covariance and correlation coefficient
  • variance of sum
  • independence of random variables, and variance of sum in this case
  • expectation of a conditioned random variable
  • expectation of a random variable as a weighted sum of expectations of conditioned random variables

Some of the questions will be testing your knowledge of basic definitions/concepts. Other questions will be problems applying these principles.

I'll have office hours as follows prior to the exam:

• Thursday, 11am to noon
• Thursday, 3pm to 4pm
• Friday, 3pm to 4pm
• Monday, 9.20am to 10.40am.

The first midterm exam will be in-class on Monday, February 25. It will be on everything we have done this semester, up to (but not including) specific named discrete probability distributions (Section 2.2 of the text). Note than this does not include material from Section 2.1 on independence of random variables, which we will return to later.

Some of the questions will be testing your knowledge of basic definitions/concepts (such as the rules of a probability function, definitions of concepts such as conditional probability and independence, and statements of principles such as law of total probability and Bayes' theorem). Other questions will be problems applying these principles.

I'll have office hours as follows prior to the exam:

• Thursday, 1pm to 2.15pm
• Friday, 3.45pm to 4.45pm
• Monday, 9.20am to 10.20am.

There will be two in-class midterm exams, tentatively scheduled for:

• Midterm 1: Monday February 25
• Midterm 2: Monday April 8

There will also be a (cumulative) final exam:

• Final exam: Tuesday May 7, 8am-10am, room TBA

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Other material

Here is where I will post any material other than homework, quizzes and exams that might be relevant for the class.

• Slides of limit theorems from the last day of classes.
• Notes of the continuous random variables.
• Notes of the functions of continuous random variables, and Gamma distributions.
• Notes of the discrete random variables.
• Notes of the ``secretary problem''.
• Slides from lectures.

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Getting help

Mathematics, like all the other sciences, is not a solitary discipline. It is a collaborative, communicative affair. Your mathematical skills will thrive by practicing talking mathematics. I encourage you to take every advantage of the opportunities available to you to do this. In particular, please contribute in class, and please come to office hours when you need to. I encourage you also to talk to each other. Share knowledge, share concerns, share questions.

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Conduct

Honor code: You have all taken the Honor Code pledge, to not participate in or tolerate academic dishonesty. For this course, that means that although you may (and should) discuss assignments with your colleagues, you must write the final version of each of your assignments on your own; if you use any external sources to assist you (such as other textbooks, computer programmes, etc.), you should cite them clearly; your work on the mid-semester exam and the final exam should be your own; and you will adhere to all announced exam and class policies.

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