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)

Test 3 Extra Credit

As promised, here is the extra credit I promised in class. Problems are based on some of the commonly missed problems from the test. The points will go toward your overall test grade.

The extra credit is due on March 31 at 10:15 AM. Late submissions will not be accepted.

Download (PDF, 53KB)

Here’s the link to download: https://math114barnard.wordpress.ncsu.edu/files/2016/03/Extra-Credit-Test-3.pdf

Study Guide For Test 3

For your upcoming test:

  • 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!

 

Extra Credit!

I know all of you are thinking: Man! I’m totally going to miss finite math and all of these counting problems over spring break!

Well never fear! Here are some additional practice problems. Hand in these problems 5 minutes before class on Tuesday March 15 (or via email) to receive up to 10 extra credit points toward your quiz grade.

  1. (5 points) Verify that the formula $latex \binom{n}{r}$ satisfies Pascal’s Indentity. In your solution, do not plug in numbers for n and r. (Hint: A small example may be helpful–check out your lecture notes from Tuesday March 1.)

For the next two problems you will use the Binomial Theorem which says:

$latex (x+y)^n = C(n,0)x^n y^0 + C(n,1) x^{n-1}y^1 + C(n,2) x^{n-2}y^2 + \ldots + C(n,n) x^0 y^n$

So for example, $latex (x+y)^3 = x^3y^0 + 3x^2y + 3xy^2 + y^3$.

  1. (2.5 points) Use the Binomial Theorem to show that the number of all subsets of the set {1,2,…,n} is $latex 2^n$. (Hint: problem number 5 from the practice problems for Pascal’s Triangle may be helpful.)
  2. (2.5 points) Use the Binomial Theorem to show that the alternating sum of each row of Pascal’s Triangle is zero. (Hint: problem number 6 from the practice problems for Pascal’s Triangle may be helpful.)