## Final Exam Study Guide

### Featured

Here’s an assortment of problems from the sections that will be covered on the final exam.

You can use these problems to make a practice test. Make sure you also review test problems from the previous exams for material not included in the book chapters (like the Simplex method and Pascal’s triangle).

Chapter 1

• Section 1.2: 15, 17, 19, 39, 41, 43
• Section 1.3: 1,3,5, 9, 11

Chapter 2

• Section 2.1: 39, 41, 43, 49, 51
• Section 2.2: 17, 19, 21, 25, 27, 29, 37, 39
• Section 2.3: 33-49 (odds)
• Section 2.4: 1, 19, 21, 25
• Section 2.5: 1-9 (odds)

Chapter 3

• Section 3.1: 7, 10, 11, 13
• Section 3.2: 25-31 (odds), 33-41 (odds)
• Section 3.3: 22 (solution: operate refinery I for 4 days and refinery II for 3 days), 23 (see example 1 for a similar problem), 24, 25, 26 (solution: ship 4 cars from Baltimore to Philadelphia, 1 car from Baltimore to Trenton, and 6 cars from NY to Trenton), 31
• Make sure to review the Simplex method as it was presented in class
• Make sure to review finding vertices of higher dimensional feasible sets

Chapter 5

• Section 5.1: 1-13 odds only, 21-26, and 41-47 (do both even and odds for the last ten problems)
• Section 5.2: 15-38 odds only (here are examples where you are shading in a region on a Venn Diagram)
• Section 5.3: 11-19 (odds only), 26-30, 61-68 (these last four questions are similar to the flag example we did in class)
• Solutions for Section 5.3:
• solution to problem 26: 102,
• solution to problem 28: 52
• solution to problem 30: 68
• solution to problem 62: 36
• solution to problem 64: 56
• solution to problem 68: 40
• Section 5.4: 1-21 odds only, 37, 38 (The solution for number 38 is 2^5), 49
• Section 5.5: 21-29 odds, 29, 31, 34 (solution: 35*34*33*32*31),35, 37, 39, 49, 51, 61, 63
• Section 5.6: 1,3, 7, 9, 11, 13, 21
• Be sure to review Pascal’s Identity and Pascal’s Triangle!

Chapter 6

• Section 6.1: 1, 3, 11, 13, 15, 17, 19, 23
• Section 6.2: 11, 13, 19, 23, 25, 29, 31, 33, 35
• Section 6.3: 7, 9,11, 17 (Outcomes are recorded as sequence (a,b,c), in which a, b, and c are days of the week), 21, 25, 27, 29, 33
• Section 6.4: 1, 3, 5, 7, 9, 11, 13, 25, 33, 35, 37, 39, 43, 45, Challenge: 63
• Section 6.5: 1, 3, 5, 7, 9, 11, 13, 17, 25

Chapter 8

• Section 8.1: 7-17 (odds only), 19, 21, 23, 25
• Section 8.2: 7-15 odds only

# Extra Solutions

Here you’ll find an alternative solution to the card problem from our Tuesday review session. I’ve also included a solution to number 9 from Section 6.4.

# Study Guide Test 4

Here is a study check list for test 4:

• Definition of Experiment, Sample Space, Event, and Outcome
• Definitions of the intersection, union and complement of events
• Given an events $latex E$ and $latex F$, you should be able to explain in words what the event $latex E\cup F$, $latex E\cap F$ and $latex E’$ (See Section 6.1 homework)
• Know when two events are mutually exclusive.
• Know the definition of a probability distribution. (See Section 6.2 for lots of examples.)
• Know how to compute probability of an event, given that all outcomes are equally likely (See section 6.3)
• Know the inclusion-exclusion rule for probability: $latex Pr(E\cup F) = Pr(E) + Pr(F) – Pr( E\cap F)$ (See Example 11 from Section 6.2.)
• Use inclusion-exclusion to fill in a Venn Diagram (For some good problems see number 33 and 35 from Section 6.2 and numbers 1-8 from Section 6.4.)
• Know the complement rule: $latex Pr(E) = 1-Pr(E’)$
• Use the complement rule to compute probability
• Good clue to look for: `at least 1’…. See for example, the birthday example (example 6 in section 6.3)
• Know how to compute conditional probability (See Section 6.4)
• Know the definition of independence: Two events are independent if $latex Pr(E\cap F) = Pr(E)\cdot Pr(F)$ (See Section 6.4, problems 33-45)

# Study Guide For Test 3

• Know how to verbally and mathematically describe a set
• Know the definition and how to use unions, intersections, and complements
• Know how to represent set(s) with a Venn diagram
• Know how to shade the appropriate region of a Venn diagram corresponding to a given set
• Know how to use the principle of inclusion-exclusion for counting
• Know how to find the number of objects in a each of the non-overlapping regions of a circle Venn diagram
• Know how and when to use multiplication in counting
• Know the definition of C(n,k), and its formula
• Know Pascal’s Identity: C(n,k) = C(n-1,k) + C(n-1,k-1)
• C(n,k) counts k-element subsets of the the set {1,2,..,n}. Which of these subsets are counted by C(n-1,k-1) in Pascal’s Identity.
• You should be able to use Pascal’s triangle to compute various C(n,k)
• Know when a counting problem involves counting subsets
• Examples:
• Selecting balls and putting them in a box
• Handshakes
• Number of matches in a league
• Many many more!
• n coin tosses with exactly k heads (see Section 5.6)
• Routes through a city (see Section 5.6)
• When counting routes in a city (see Section 5.6): Given a route you should be able to write down the subset that corresponds to it.

In general, the best thing to do to study for this test is to review the examples from lecture and homework. The more practice you have, the better you’ll be able to handle the questions on the test!

# Test 2 Study Guide

• Know the steps for solving linear programming problems in two variables
1. Find constraints (this is a list of inequalities) and objective function
2. Graph feasible set. (Suggestion: Find the x and y intercepts of each constraint line), and list the vertices of the feasible set.
3. Test each vertex in the objective function until you find the optimum (minimum or maximum) vertex.
• Be able to carry out each of the steps above
• For applicable linear programming problems: Be able to eliminate all but two variable by using an equation from the word problem (As in the shipping example we did recently.)
• Be able to explain why an interior point in the feasible set is never optimal
• Know how you can find a vertex of the feasible set when it has three variables
• Be able to solve a linear system, and verify the solution (the intersection point) is feasible or not feasible
• Know the steps involved in the Simplex Method
• Given a recipe for the set of vertices of a feasible set, a recipe for finding the neighbors of a vertex, the objective function and a starting vertex, you should be able to carry out the Simplex Method
• Know the definition of a set and a subset
• Be able to explain the difference between a set and a list
• Be able to list all of the subsets of a given set