Final Exam
Review I
Chapters 2-4
General Information,
Objectives, and Examples
Final Exam – General Info
 Saturday, Dec 13,1:00-3:00 p.m. Location posted in Oncourse (arrive
by 12:45 p.m.) ― more info on final exam day is available via Oncourse
 Bring your JagTag card, a few sharpened pencils, a ruler, and your TI30Xa calculator to the exam.
 There will be 23 numbered questions on the exam (although several will
have multiple parts to them).
 The questions will be in chronological order from Ch. 2 to Ch. 9.
 Every question will have a point value listed. The test contains a total
of 100 points. The final score will be scaled to150 points.
 The questions will be open response.
 Partial credit will be awarded where it’s warranted – so you should
show all your work to fully justify your answers. Clearly mark your final
answers. Answers that are not legible will be marked wrong.
Final Exam – How to Prepare
 There are three important sets of problems that I
recommend working to best prepare for the final exam:
1) Your previous tests
2) The practice final exam (under the “Final Exam
Resources” folder in Oncourse)
3) Old practice tests reopened on Maple
 While working the problems above, use homework
assignments and book problems to supplement
problems/concepts that you need more practice with.
Chapter 2 – Final Exam Objectives
 Given sets, perform set operations to either form a new set or to determine
the new set’s cardinality.
 The operators are:
“Union”
“Intersection”
“Complement”
“Relative Complement”
“Cartesian Product”


′


Combining the elements of two sets (without repeats)
The elements that two sets have in common.
The elements that are missing from the set
(that exist in the Universal).
A – B would be the elements that are in A, but not in B.
The ordered pairs obtained by pairing up elements of two sets.
 Given a set, determine the number of different subsets that can be formed.
2
n
 Given a word problem, be able to use sets and Venn Diagrams to find a solution.
 Given a set that is partitioned, find the number of elements of one of its subsets.

n(U )  n( A)  n( B)  n(C )
Know De Morgan’s Laws. A  B  ( A  B )
A  B  ( A  B )
Ch. 2 Examples
[1] Let U = {1, 2, 3, h, j, k} ; A = {1, j} ; B = {2, 3, h, j} ; C = {3, k}
(a) Find (A' ∩ B') ∩ C
(b) Find A x B'
(c) Find n[B – (A U C)]
Ch. 2 Examples
[2]
Ch. 2 Example
[3] For a survey of 100 students, it was found that:
35 took Astronomy
40 took Biology
12 took Biology but not Astronomy
(a) How many of the students took both of these classes?
(b) How many of the students took neither of these classes?
(c) How many of the students took at least one of these classes?
(d) How many of the students took exactly one of these classes?
Chapter 3 Objectives – Know how to use the three main counting methods from Ch. 3:
Section
3.1
3.2
3.3
Method
F.C.P.
Permutations
Combinations
Repetition?
Order?
Examples
Allowed
Order of
selection
matters
License plates, zip-codes,
phone numbers, SS#’s,
passwords, seating people
in a row, answer sheets for a test,
trip itineraries, constructing
numbers in a range, etc…
Not
Allowed
Order of
selection
matters
Assigning titles to people (e.g.
Pres, VP, Sec), Awarding prizes
that are different (e.g. 1st, 2nd, 3rd),
and some of the F.C.P. examples
above (as long as repetition is
not allowed), etc…
Order of
selection
does not
matter
Selecting people to form teams or
committees (without titles),
selecting marbles from an urn, card
problems, awarding prizes (if equal),
creating subsets that have a specific
number of elements, handshake
problems C(n,2), etc…
Not
Allowed
Chapter 3 – Final Exam Objectives (cont.)
 Given an experiment, draw a tree diagram to find the number of possible
outcomes
 In this section trees were used mainly for experiments where the
number of stages could vary.
Example: A box contains 2 green, 1 white, and 1 purple marble. An experiment consists
of drawing marbles in succession without replacement, and noting the color of each until
a white marble is drawn. How many outcomes are in the sample space?
Ch. 3 Examples
[1]
How many 5-digit codes are possible if the first two spots must be
different letters of the alphabet and the last three spots can contain
any number from the set (1,2,3,4,5,6)?
[2]
There are 6 people in a race. How many different ways can these 6
people finish the race?
Ch. 3 Examples (cont.)
[3]
In a jar there are: 10 red jelly beans,
7 yellow jelly beans
4 green jelly beans.
Five jelly beans are randomly selected.
How many different ways can this be done if:
(a) It must include exactly 1 red?
(b) It must include the same color?
(c) It must include at least 1 red?
Chapter 4 Objectives
 Given an experiment with equally likely outcomes, compute the probability of an
event by using the basic probability formula (4.1):
n[ E ]
Pr[ E ] 
n[ S ]
 Know how to apply fundamental properties of probability (4.1 and 4.2):
Complement Formula:
0 ≤ Pr[E] ≤ 1
1 – Pr[E′] = Pr[E]
Union Formula:
Pr[A U B] = Pr[A] + Pr[B] – Pr[A ∩ B]
For any event E:
For Disjoint events:
For Independent events:
Pr[A ∩ B] = 0
Pr[A U B] = Pr[A] + Pr[B]
Pr[A ∩ B] = Pr[A]∙Pr[B]
Pr[A U B] = Pr[A] + Pr[B] – Pr[A]∙Pr[B]
Chapter 4 Objectives (cont.)
 Given an experiment, determine the conditional probability of an event (4.2 and 4.3):
n[ A  B]
Pr[ A B] 
n[ B]
which can also be expressed as:
Pr[ A  B]
Pr[ B]
 Be able to construct a probability tree, when appropriate, and use the tree to answer
various questions regarding probability.
 Be able to construct a probability Venn, when appropriate, and use the Venn to
answer various questions regarding probability.
 Use the binomial probability formula for experiments that involve a sequence of
nk
Bernoulli trials (4.4)
k
C (n, k )( p ) 1  p 
Ch. 4 Examples
[1]
A bucket contains 10 light bulbs: 7 are blue and 3 are red
Three bulbs are selected at random.
(a)
What is the probability that exactly 2 blue are selected?
(b) What is the probability that they are all blue given that they are all the
same color?
Ch. 4 Examples (cont.)
[2]
Vince will go to the football game with a probability of 0.70, and he will go to
the opera with a probability of 0.10. Assuming these events are independent:
(a) What is the probability that he goes to at least one of these events?
(b) What is the probability that he goes to the football game, but not the
opera?
Ch. 4 Examples (cont.)
[3]
An unfair coin with Pr[Tails] = 0.35 is flipped six times.
(a) What is the probability of getting exactly 2 tails in the 6 rolls?
(b) What is the probability of getting at least 1 tail in the 6 rolls?
Ch. 4 Examples (cont.)
[4] At a local factory, 85% of the workers are male. Of the male workers, 70%
were born in-state. Only 55% of the female workers were born in-state.
If a worker is randomly selected from the factory:
(a)
Find the probability that the worker is an out-of-state male?
(b) Find the probability that the worker was born out-of-state, given that the
worker is a female?
(c) Find the probability that the worker is a female, given that the worker was
born out-of-state?