From section 2.6:

Test 1 is scheduled for the first hour or so of class on Wednesday. Please make sure that you are familiar with the course policies.

The Test 1 review problems which were handed out in class are also available here.

Here are the answers and some partial solutions and hints (corrected for problem 4d):

MAT2572Test1ReviewAnswersSpring2018

Here is the outline of Probability rules (theorems) that I handed out in class previously:

MAT2572ProbabilityRules

Here are the notes on Conditional Probability and Independence:

Math2501ConditionalProbabilityIndependence

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

(Notes soon)

• Review the examples of Bayes’ Theorem we discussed in class, and also the definitions of independence.

• Answer the questions on page 1 of the handout “Conditional Probabilities and Contingency Tables”: here is a much improved version which corrects some bad references and improves the second page problem (and explains the notation) – please use this version!

• Also answer the following:

For problem 2.4.25, if we find that the ball bearings are defective, what is the probability that they came from Company 3?

• Do these problems from the textbook:

p. 52 #2.4.45

p. 58 # 2.5.1, 2.5.3, 2.5.7, 2.5.9

Also, if you have not yet joined our Datacamp group, please check your email for the invitation to join our Datacamp group. The email was sent to whatever email address is your main address in CUNYFirst.  It is important that you join the group as soon as possible, as there may be a deadline. Also, you must join using the email to which your invitation was sent. I believe that you can also go directly to Datacamp and create an account using that email, and then you should see a notification in Datacamp that you have been invited. The notifications are under the little icon shaped like a bell on the upper right side of the webpage.

There will not be a Quiz on Wednesday.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Please note, there are no classes on Monday the 19th because of Presidents Day. On Tuesday we follow a Monday schedule, so this class will meet.

I hope to soon post more notes about what I did on Wednesday (and other things). In the meantime, here is a summary of the various formulas and theorems about Probabilities that we have so far, plus a few more that are coming up next time:

MAT2572ProbabilityRules

I have posted a copy of the homework problems on Piazza. You should do problems #2.4.25, 2.4.26, and 2.4.31. Please bring the handout with the contingency table problems again next time: you can try to work the problems yourself in the meantime of course!

Also please check your email for the invitation to join our Datacamp group. The email was sent to whatever email address is your main address in CUNYFirst. For most of the people who have not yet joined the group, that would be your citytech email address. It is important that you join the group as soon as possible, as there may be a deadline. Also, you must join using the email to which your invitation was sent. I believe that you can also go directly to Datacamp and create an account using that email, and then you should see a notification in Datacamp that you have been invited.

There will not be a Quiz on Tuesday.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Open to all. It looks interesting!

Title: “Let’s Play Sudoku”

Speaker: Brad Isaacson (NYCCT)

Date/Room: Thursday Feb. 8, 2018, 12:50-2:00pm, Namm N720

Abstract:

Sudoku is a very popular number puzzle.  In this talk, we will discuss some of the strategies for playing Sudoku, including naked and hidden pairs/triples, unique rectangles, and alternating inference chains.  May there never again be a Sudoku puzzle that you are unable to solve.

Pizza and refreshments will be served at 12:45pm.

Topics:

• More on probability distribution functions on sample spaces:

As a consequence of the axioms and theorems from last time, We can (and usually will) define a probability distribution on a sample space S by assigning a probability to each outcome in S, in such a way that every probability is non-negative and they all add up to 1 (= the probability of S).

Then we have the Theorem:

For any event A in S, P(A) is the sum of the probabilities of the outcomes in A.

• Conditional probability

The notes I showed on the screen in class are here:

Math2501ConditionalProbabilityIndependence

We only went through basically the first page of those notes. I’ll write more notes on the other examples here when I have time.

Homework:

• Be on the lookout for an email invitation to join our Datacamp group, coming soon to whatever email address Blackboard has for you!

• If you have not already joined Piazza: Look for the invitation to join our Piazza discussion board, and join by following the link. The invitation has been sent to whatever email address you have in Blackboard, and it will be from “The Piazza Team”. Or you can go directly to Piazza here and join using your City Tech email address. We will be using the discussion board as a way to ask questions about the homework or the material discussed in class, and also you will be submitting some of your R assignments via Piazza.

• Do the following from the textbook: (I have posted a copy of these problems over on Piazza on the “Resources” tab)

• Monday 12 February is a holiday (no classes meet).

• There will be a Quiz on Wednesday. The question will be a randomly selected homework problem from today’s assignments.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Note on my schedule: for the time being, I will only be able to be online to read and reply to emails at certain times of the day. (It is possible that I may be online at other times but I cannot guarantee it.) The times are roughly:

Monday – Friday early morning

Monday-Thursday around 2:30-3:00 PM

Sunday-Thursday evenings around 9-10 PM

Please be aware of this if you need to contact me by email. Thanks.

Topics:

• The three types of sample spaces (according to how many outcomes they contain)

These are:

Two types of discrete sample spaces:

a finite sample space – contains a finite number of outcomes

an infinite discrete, or countably infinite, sample space – contains an infinite number of outcomes which are countable in the sense that they can be put in a one-to-one correspondence with the natural numbers.

One type which is not discrete:

a continuous sample space –  for us usually consists of a (possibly infinite) interval on the real line or a region in the plane – or could be in more dimensions. What distinguishes this type is that there is an uncountable number of outcomes.

These types of sample spaces will require different ways of defining probabilities and different methods of working with probabilities. You will see that in infinite sample spaces, whether they are discrete or continuous, some rather strange things can happen.

• Defining a probability distribution on a sample space. (For now, we are just treating the probabilities as numbers which satisfy the axioms given below, without worrying about what they mean.)

Definition: For a finite sample space S, a probability distribution on S assigns a real number P(A) to every event A in S, so that the following axioms hold:

Axioms:

1. For every event A in S, $P(A) \ge 0$
2. $P(S) = 1$
3. If A and B are mutually exclusive events in S, $P(A\cup B) = P(A) + P(B)$

[If S is a countably infinite sample space, we will have to add another axiom to this list. For continuous sample spaces we will have to make some changes!]

A number of theorems follow from these axioms: (Proofs were done for some of these in class, and you can read them in the textbook. I’ve given hints for how to prove them below.)

Note: I will be using the prime notation for complements in these notes.

Theorem: $P(A’) = 1 – P(A)$

The proof comes from the fact that $A$ and $A’$ are mutually exclusive and their union is the whole sample space.

Theorem: $P(\varnothing) = 0$

You might think this is obvious, but it does have to be proved! The proof comes from the fact that the null set is the complement of S, and we use the previous theorem.

Theorem: If $A \subset B$, then $P(A) \le P(B)$

The proof comes from breaking down B into two mutually exclusive pieces: $B = A\cup (B\cap A’)$. (Draw a Venn diagram to see why this is true.)

Theorem: For any event A, $P(A) \le 1$

The proof comes from the fact that all probabilities are non-negative (axiom 1) together with the fact that $P(A) + P(A’) = 1$ (which was used in proving the first theorem).

This theorem is very important to keep in mind. It often happens that people make errors in computing probabilities and end up with a probability which is larger than 1. You should instantly realize that this must be wrong. (Similarly if you end up with a probability which is negative.)

Theorem: For a finite collection of events $A_1$, $A_2$, … , $A_n$ which are pairwise mutually exclusive (that is, every possible pair of them are mutually exclusive),

$P(A_1 \cup A_2 \cup \dots \cup A_n)  = P(A_1) + P(A_2) + \cdots + P(A_n)$

In shorthand, this is written as

\[ P\left(\bigcup_{i=1}^n A_i\right) = \sum_{i=1}^n P(A_i)\]

Consequence of all the above: Especially considering the theorem just above, this means that we can define a probability distribution on a finite sample space by just giving its values for the outcomes in the sample space, and this is how we will do it in the future. The probabilities of the outcomes can be chosen in any way at all as long as they are all non-negative and they add up to 1.

Theorem: $P(A\cup B = P(A) + P(B) – P(A\cap B)$

Look at the Venn diagram to see why this is true.

Examples discussed in class: From the textbook, Example 2.3.3, and problems 2.3.1 and 2.3.4

Please note that the problems intend you to use the axioms and theorems given above, plus some very basic ideas of where probabilities come from. Several problems also assume that you will use “equally likely outcomes”, for instance in Example 2.3.3 they assume that because there are 3 cards of the same rank as the card you drew first, and 51 total cards left in the deck, that you can conclude that the probability of drawing a card of the same rank is 3/51. I want to point out that they have not justified defining a probability this way yet, and don’t get too used to it, because it needs to be justified! In particular, Example 2.3.3 should have specified that the cards were being chosen at random, or some other wording that would mean that every card was equally likely to be chosen.

I’ll try to write up my discussion of the examples but it may not happen today!

Homework:

• Look for your invitation to join the Piazza discussion board for this class, which should have been sent to whatever email address Blackboard has for you. It would be from “The Piazza Team”. (New students, it will be sent sometime on Tuesday.) Or you can sign up by going directly to Piazza and use your City Tech email address to verify you are a student here.

• The homework problems and information about the quiz next time are posted here.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Note on my schedule: for the time being, I will only be able to be online to read and reply to emails at certain times of the day. (It is possible that I may be online at other times but I cannot guarantee it.) The times are roughly:

Monday – Friday early morning

Monday-Thursday around 2:30-3:00 PM

Sunday-Thursday evenings around 9-10 PM

Please be aware of this if you need to contact me by email. Thanks.

My post for Monday’s class is delayed as I am arranging things computer-wise, so in the meantime I am posting these problems. From the pages below, the assignment is:

# 2.3.2, 2.3.5, 2.3.9, 2.3.10, 2.3.15, 2.3.16

MAT2572section2point3assignment

Also, there will be a quiz on Wednesday. The quiz will consist of one of the questions from the first day homework. It will be “open-book” in the sense that you will be allowed to use your notes during the quiz.

In future the quiz will usually consist of a question from the homework that is due on that same day, so be sure to make use of Piazza  to discuss problems before they are due!

This site is where you will find all the information, notes, links, assignments, and so on for this class. Please check back frequently!

I am trying to make a visualization of the proof of one of DeMorgan’s Laws that I did in class using Venn diagrams, but it’s taking longer than expected. So in the meantime, here are the notes that I showed in class:

Math2501SetNotation

Also please read Section 2.2 in the textbook.

Homework:

• Get the textbook and the materials that you will need, especially the flash drive or cloud storage for your work files in R. The textbook is:
An Introduction to Mathematical Statistics and Its Applications, 5th edition, Larson and Marx (Pearson publishing)

• Review the material about sets and set operations, and application to sample spaces, that we discussed in class. Here is some important vocabulary:

Experiment

Outcome

Sample space

Event

• Do the following problems from the textbook: I may ask for some of them to be put on the board next time.

p. 20 #2.2.1, 2.2.4, 2.2.5, 2.2.7, 2.2.10

• Read in Section 2.2 pp. 21-23 (which gives examples of unions, intersections, and complements in actual sample spaces). Then work the problems on pp. 24-26 #2.2.21, 2.2.22, 2.2.29, 2.2.37. We will discuss them next time.

In case you are not able to get the textbook before next time, here are pages 20-26 which contain the above assignment:

MAT2572assignmentDay1

• Also look in your email for an invitation to join our Piazza discussion board, which should arrive in the next day or so. It will be sent to whatever email address you have in CUNYFirst, most likely your City Tech email address.