Math 1312 - Business Math II
Review of Sets and Set Properties
Consider Each of the following Situations
1.
A company employs 20 women, 4 accountants, and 6 sales agents.
How many individuals are employed by the company ? _____________
How many of the accountants and sales agents are women ? _______
If a group of three is selected , what is the probability that of the three selected 2 are women ? __________
2.
An investment analyst describes 10 possible companies to invest in. Last year he was right on 90 out of 100
companies. You pick one at random. Invest in it. What is the likelihood that you picked a “good” investment ?
3. A new pill for diabetics claims to lower sugar levels to a normal level in all but 8 % of all cases. A group of 500
takes the pill to verify their claims. How many individuals should be expected to have a lower sugar level ( normal ) ?
4.
Grades are known to be normally distributed with mean 70 and variance 9.
What is the probability that the student has a grade of 80 or higher ?
5. The cost in cleaning up a river is given by g(x) =
A student is selected at random
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, where x represents the percent of purity.
100  x
What will be cost to return the river to 100 % clean ?
2. Sets
A set is a collection of numbers or objects that have a well-defined property in common.
Which of these represent well-defined sets ?
ex. Set of all smart students in class - ____________
ex. Set of all students with long hair - ___________
ex. Set of all good fruit - _______________
ex. The set of all students enrolled in math 1312.050 on Jan. 25 during the spring semester of 2005.
There are times that we can list all members of a set - { .... }
{ a, e, i, o, u }
{ A, B, C, D, F , W}
{ 2, 4, 6, 8 }
But other times we can not:
although by listing several members – we get the idea of what the set contains
{ 1, 3, 5, 7, ... },
{ red, blue, gray, purple, green, violet,... }
{ Joe, Jim, John, Jeff, Jack,... }
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We can use what’s is called set-builder notation.
This way we do not write every member of a set – just indicate what the members look like.
{ x | x represents a student enrolled in math 1312.050 – spring 2005 on Jan. 25}
{ x | x represents a brand of car that you have driven }
{ x |x represents the amount of money in a students bank account }
In some cases – one form may be better than the other.
{ a, e, i, o, u }
{ x | x represents a student currently enrolled at A.S.U }
The members of the set are called elements of the set.
member of a set.
capital letters: sets
We use the symbol  to express the fact that an object is a
lower case letters: elements
A = { a, b, c } or B = { 5, 10, 15 ..... } , C = { x | x is a positive real number }
We say a  A, 5  B, d
 A, or 7  B
Is 0  C ? ___________
Is there a smallest member of C ? _____________
Sets of numbers:
N = set of natural numbers = set of counting numbers = set of positive integers
= { x | x  set of natural numbers } = { 1, 2, 3, 4, .... }
W = set of whole numbers = set of nonnegative integers
= { x | x  set of whole number } = { 0, 1, 2, 3, ... }
I = set of integers =
= { x | x  set of integers } = { ... -3, -2, -1, 0, 1, 2, 3, ... }
Q = set of rational numbers
= { x | x  set of rational numbers } = { m/n | m and n are integers with n ≠ 0 }
We can not list the members of this set as nicely as we did the first three. The best we can do is
say the above statements and provide examples.
- 3 = -3/1, 0 = 0/6, 4 = 4/1, 2/3, -5/17, 0.24, 0.11111... ( 1/9 ) , ...
Q / = set of irrational numbers → examples:
π, e,
2 , 0.1010010001... , ...
R = set of real numbers : consists of the set of rational and irrational number – no other number
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Sets
Def. A set is a collection of objects having a well defined property(characteristic) in common.
- which of these can be classified as sets ?
_________
Set of all female students in this class .
____________
the set of all companies with smart chairman
___________
set of all companies with executives who were given a salary exceeding 10million dollars / year
At times it is useful to list all members of a set but when this is not feasible or desired we can also write sets in
set-builder notation
4. A = { x : x is a whole number less than four } = _____________________________
5. B = { x : x is an integer with | x | < 2 } = ____________________________
6. C = { x : x is a current Fortune 500 company } = ______________________________________
7.
D = { x : x represents the name of a company that has not had a posted a losing quarter in the last 20 years }
= _______________________________
Notice this last example. If no such company exists, then this is a set that is called an __________, or the ________
Def.1.1
A ___________ is a collection of objects having a well defined property in common.
Def. 1.2
The
set that contains all objects under consideration is called the ______________________
-----Depending on the Experiment – the universal set will vary.
ex. Find all x’s so that
A = { x | x2 = 4 }
B={x|x<0 }
C = prime numbers
Def.1.3
The objects – or members - of a set A are called _________________ of the set.
ex. If A = { 1, 2, 3 } we write 2 A and say “ ____________________________________________”
we write 5  A and say “ _____________________________________________”
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Venn Diagrams are used to illustrate sets and their relationship to each other and the universal set U. Consist of rectangles
and “circles” to represent the sets and the universal set.
The first relationship between sets:
Def. 1.4
Let A and B be any two sets. We say A is a ___________________ of B provided every element of A is also in B.
Note: If A is not a subset of B ( A
 B ) , then there exists an element in A that is not in B.
Example.
A = {x : is a natural number < 5 },B = { x : x is an even prime number }, C = {x: x is the largest negative whole number }
List the members of each set.
A = _______________________________
B = ___________________________
C = ____________________
Are any of the sets above subsets of another set listed ? ______________________
Example.
A = { set of students in class that are male },
B = { set of students in class that are over 20 years of age }
C = { set of students in class that have a daytime job }
In general, which of these sets can be classified as subsets of each other ?
What is the universal set ?
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Def.1.5
Let A and B be any two sets of some universal set U. We say that A and B are said to be ____________ provided
A  B and B  A.
ex. Is { a, a, b }  { a, b } ?
Conclude that 1.
____________
What about { a, b }  { b, a } ? ________
Repetitions do not count – if there are repetitions, we can rewrite the set with no repetitions
2. Order does not matter – order that the elements are written is not important
We want to be able to count the numbers of subsets of a given set.
ex. How many subsets does the set
{ a } have ? _________
What about the set {1, 2 } ? ____________
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Properties of sets
1. We use  to compare an element to a set
2  A is acceptable or 5  B is also acceptable but
{ 1, 2 }  A is not. Why ? ____________________
examples:
a)
A = { x | x is an so that x2 > 0 }
Is - 2  A ? _____________
Is 25  A ? ____________
Is there any integer that is not an element of A ?
b) B = set of all students in class that are enrolled in math 1312 but have not taken math 1311.
Is B empty ? ___________
2. We use  to compare a set to a set
{ 1, 2 }  A or { 1}  { 1, 2 }
or the set of natural numbers is a subset of the set of integers , N  I
but we can not use 2  A. Why ? _________________________________________
ex. List some of the subsets of the set A = set of all letters of the alphabet that are considered vowels.
ex. True or False.
0  set of whole numbers . ___________________
3. The universal set U contains all objects under consideration and the empty set (the null set ) contains no object at all
U - the universal set ,
 or { } - the null set, the empty set → we do not use {  } to represent the empty set
example: let A = { x | x 2 = 4 }
example: let the set B represent the students that voted in the last local elections.
4.   A for any set A , the null set is a subset of any set including itself (    )
for any set A, A  A.
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Def.1.6
Let A be a subset of B. If B is not a subset of A, then A is not equal to B and A is said to be a proper subset of B.
ex. Consider the set of all individuals in this classroom ( set A ) and the set of students in this classroom (B ) . How are
these two sets related ?
Are they subsets of each other or are they proper subsets ?
ex. Consider the set of all male students in class today ( M ) and the set of all students in class still awake and listening ( A ).
How are these two sets related ?
ex. Let U = set of all integers less than 10 , A = { x | x2 = 16 } , and B = { x |
Which of these statements is true ? A  B
BA
16
= x }
B =A
Are either A or B proper subsets of each other ?
Counting subsets:
How many subsets does the set B = { even prime numbers }
How many subsets does the set B = { 1, 2 } have ? ___________
How many subsets does the set A = { a, b, c } have ? _________
How many subsets does the set A = { a, b, c, d have ? _________
In general how many subsets does a set with n objects have ? _____________ how many proper subsets ? ________
A committee is to be appointed from a group of five.
The committee can consist of 1 , 2, 3, 4, or 5 individuals. How many distinct
committees are possible ?
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Def.1.7 (Disjoint)
Let A and B be any two nonempty sets. If A and B have no element in common ,
then A and B are said to be
______________ .
Use Venn Diagrams to illustrate sets that are
disjoint sets
sets that are not disjoint
subsets of each other
ex. Which of the following pairs of sets are disjoint ?
__________________
1) A = { x : x is an even natural number }
__________________
2) C = set of all nonnegative integers
___________________
B = { x : x is an odd whole number }
D = set of all non-positive integers
3) set of all
ex Let C = { x : x is a rational number },
F = { x : x a nonnegative integer }
D = { x : x is an integer }
E = { x : x is a whole number }
G = { x : x is a natural number }
Are any of these sets equal ?
Which is the “largest” set ? Is there one ? ( largest in the sense that it contains all of the other sets )
Def.1.8 ( Complement )
Let A be any set of some universal set U. We define the complement of A, A/ as the set that contains
all objects in U that are not in A.
ex.
ex. Let U = { x : x is a whole number } with A = { x : x is positive } ,
Find A/ = ____________________________
B = { x : x is even whole number }
B / = ______________________________
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Let A = set of all days in which rain of 1 inch or less fell in San Angelo.
A/ =
Let B = set of all students in class with at least one ring.
B/=
Let C = set of all students that are at least 40 years of age or older
C / = ....
Let D = set of all four card hands with at least one diamond.
D/ =
Def. 1.9(  and  )
Let A and B be any two sets of some universal set U. We define
1) the intersection of A and B (written A  B ) as
A  B = set of all objects (elements) that are in A and at the same time they are also in B
--- each element in A∩B must be classified as being part of A and at the same time part of B
ex. a couple’s mutual friends
ex.
2) the union of A and B ( written A  B) as
A  B = set of all elements that are in A, in B, or in both A and B (either A or B )
( they are in at least one of the two sets but not necessarily in both sets – although they can be)
ex. when a couple gets married and they bring in all their property.
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ex.
Let U = { all positive integers less than 5 }, A = { x : x 2 = 4 }, B = { x : x < 3 }, C = { x : x > 2 }
1) union: What is A  C = ___________
2) intersection:
A  B = ___________,
B  C = ______________
A  C = ____________ ,
A  B  C = ___________
B∩C
More properties of sets.
a. complements
2)  / = _____________
1) (A/ ) / = _________________
3) U / = ___________
b.
1) ( A  A/ ) = _________
c.
2) ( A  A/ ) = ________
De Morgan’s Law:
1) ( A  B ) / = A/  B/
2) ( A  B ) / = A/  B/
d. Commutative, associative, and distributive
1)
A u B = B u A
and
A n B = B n A
2) ( A u B ) u C = A u ( B u C )
and
3) A n ( B u C ) = ( A n B ) u ( A n C )
( A n B) n C = A n ( B n C )
and .......
Any of these laws can be proved by using Venn-diagrams.
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Review of Sets:
1. The set that contains all elements under consideration is called the ________________________ set.
2. The sets A and B are said to be ________________________ provided A ∩ B = 
3. If x  A and x  B, then x  _________ and x  ___________
( either A or B)
( both A and B )
4. The ____________ of sets A and B consists of all objects that are in A or in B or in both A and B.
5. If A = { 1, 2, 3 } and B = { 2, 3, 4 }, then another name for the set { 2, 3 } would be ____________
6. If the universal set U = { 1, 2, 3, ..., 10 } and A = { x | x2 ≥ 4 }, then the complement of A = A / = ______________
7. Find the union of A = {a, e, i , o, u } and B = { b, c, d, f, g, h, .... } in set builder notation.
8. Draw a Venn Diagram for each of the following sets. Make sure to include the universal set.
a) A  B/ =
b) A ∩ ( B ∩ C / )
9. Prove or Disprove by shading the sets that correspond to each side and indicating whether they are equal or not.
A  B / = ( A/ ∩ B / )  A . Provide your answer (work) on the back of this page.
10. True or False.
______________ a) - 2  { x | x2 = 4 }
______________ b) 0  { x | x is a whole number }
______________ c)   { 1, 2 }
______________ d )   A, for any set A
_______________e) ( A  B ) / = A / ∩ B /
______________ f) A  A, for any set A
_______________ g) U / = 
______________ h ) A  A/ = 
11. How many subsets does the set { a, e, i, o, u } have ? ____________ How many proper subsets ? __________
12. Two sets A and B are said to be equal provided 1) _____________________ and 2) ________________
13. Find the absolute value of each of the following
a) ___________________ =
14. A = { x | x2 = 4 } and
b)
= ______________
c)
=
B ={x| 1< x< 3 }
a) Can A be a subset of B ? Why or Why not ?
b) Can B be a subset of A ? Why or Why not ?
c) Could they be equal ? Why or Why not ?
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Sets and Counting(basic notation)
ex.
In a group of 100 people: 28 are known to smoke, 12 are known to have children, 42 are known to live on campus,
How many do not smoke ? __________
do not have children ?______________,
smoke and live off campus ? _____________
ex.
Let A = { x | x2 < 4, x a whole number } = ___________________
, B = {x | x is a whole number less than 10 } = _______________________
C = set of all even prime numbers = _______________
n(A) represents the number of objects – the number of elements – in set A.
n(A) = _________,
n(B) = _________,
n(A u B ) = ___________,
n(C ) = ___________,
n(A n B ) = ___________
ex. Let A = set of all students that took math 1311 last semester and B = set of all off campus students
with U = set of all students in a particular section(class) of Math 1312
Construct a Venn Diagram using the following information;
28 took math 1311 last semester,
20 are off campus students,
There are 38 students enrolled in class.
a) n(A ) = _________
18 off campus students that also took math 1311
b) n(B ) = _________
c) How many can be classified as being in both sets ? n( A ∩ B ) = __________
d) How many can be classified as being in at least one of the sets ( one or the other ) ? n(A  B ) = _______
e) How many can be classified as being part of one set but not the other ? ___________
f) How many different regions are there in the Venn Diagram ? ______________
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A student council is to be made up of willing members of the student group consisting of
1 senior, 1 junior, and 1 sophomore, 1 freshmen, and 1 Jr High student ( 6 th – 8th )
There are 4 seniors willing to serve, 8 Juniors, 6 Sophomores,
How many different (distinct ) groups are possible ?
10 Freshmen, and 7 Jr. High students
___________________________
Cross Product(Cartesian Product )
The cross product of A and B, written A x B is the following set.
A x B = { (a, b ) | a is an element of A and b is an element of B }
ex. Let A = { a, b } and D = { 2, 3, 4 } . Find
Tree Diagram:
A x D = { ______________________________ }, D x A = { _____________________________________ }
n ( A x D ) = _______________ n ( D x A ) = ___________
Notice that B x A = { (b, a ) | ....... }
So, A x B has the same number of elements as B x A but they are not equal
for example: if you plot the points (2, 3) and ( 3, 2 ), you get two different points so ( a,b ) ≠ (b, a )
We can extend the Cartesian product to more than two sets
ex. Let A = { a, b }, B = { 1, 2, 3 }, and C = { e, f } .
Then A x B x C = _______________________________________________________
Notice that n(A x B ) = __________________ also n( A x A x A xA ...) = _____________
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ex. A class consists of 4 boys and 5 girls. One pair of children is selected consisting one boy and one girl. How many ways
can such a pair be selected ? ( How many distinct pairs are possible ? – keep in mind that only one pair is being selected)
Solution:
Let the set of Boys be represented by B = { b1, b2, b3, b4 } and the set of Girls be represented by G = { g1, g2, g3, g4, g5 }
ex. A ID Card is to contain: sex(m, f), department(1, 2, 3, 4 ), seniority( 0, 1, 2 ) . How many distinct ID cards are possible?
Solution:
Let A = { m, f ), B = { 1, 2, 3, 4 }, and C = { 0, 1, 2 } .
ex. A meal is planned:
one of four desserts, one of three meats, and one of two types of salads.
How many different meals are possible ?
_______________
ex. How many distinct phone numbers are possible of the form xxx-xxx-xxxx ,if we place no restriction on the digits ?
_________________
ex. How many different soc. sec. #’s ? xxx-xx-xxxx __________________
Properties of A x B
1. A x B does not necessarily equal B x A .
2. The set A x B consists of pairs of objects – the elements are pairs -- ( 2, 3 ), ( a, 2 ), ( b, c ) -2. n( A x B ) = n(A)  n ( B ) which is the same as n(B x A ), so n(A x B ) = n(B x A )
3. We can extend #2 to more than two sets n(A1 x A2 xA3 .... ) = n( A1 )  n(A2 )  n(A3 ) .....
4) the elements of A x B x C consists of triples – ( a, b, c ),
the elements of A1 x A2 x A3 x A4 consists of 4tuples –(a,b,c,d)
Note: A x B = is a set = { (a, b ).... }
while n(A x B ) is a number
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More Review over Sets: Cross Products and Counting
1. Let A = { a, b, c, d } and B = { 1, 5 }. Find A x B.
2. If n( A ) = 5 and n(B ) = 4, then find n( A x B x A ) = __________
3. True or False.
___________ a) A x B = B x A .
__________ b) n( A x B ) = n (A ) n(B )
4.
There are four doors to exit a building. Once outside a person can ride a taxi, a bus, or a subway train,
He/She will be dropped at the destination. There are two ways to go in and three different elevators can be used to
reach the apartment. How many different routes are possible ?
5.
A game consists or rolling a die and recording the outcome. The game continues with a toss of two coins and
recording the number of heads. How many different sequence of numbers are possible ?
6.
A couple is to be selected as Prom King and Queen. Everybody is eligible for their respective position.
There are 20 young men and 24 ladies. One pair is to be selected – how distinct possibilities are there ?
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Basic Probability
Find the value of x if
x + ¼ + 2/3 = 1. x = _________________
Three tests are equally weighed so that their sum is 1 ( 100 % ). What is the weight of each one as a fraction ?
Six numbers are equally weighted so that their sum is 1. What must the weight of each number be ? _________
Consider the following experiments:
1) a coin is tossed: what is the likelihood that a head comes up if each side is equally likely to come occur ? _______
2) a family of three children is picked at random. What is the likelihood that all three children are boys ? _________
3) a fair die is tossed what is the likelihood that a number greater than four comes up ? ________
4) You wait at a bus stop. You define a bus being on time if the bus comes within 5 minutes of its scheduled stop.
What is the likelihood that the bus will be on time (within five minutes ) ? ________
5) A class consists of 20 male students and 15 female students. A student is chosen at random. What is the likelihood that
the student selected will be female ? ______
6)
Fifty-two students make up a business class. Twenty- six are management and twenty-six are finance.
group is male. A person is selected at random. What is the likelihood that the person selected is
a female student ? ___________________
Half of each
a male-management student ? ______________
8) What do you think the likelihood of guessing you birth month correctly ? ____________
What if I have two tries ? _____________
What if I have six tries ? ______________
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Theoretical Probability – vs – empirical probability
1)theoretical probability:
by assumption or by knowing the entire population
ex. you roll a die – what is the probability that a four comes up --ex. There are 50 students in class. One is selected at random. What is the probability that the student selected
graduated from H.S. in May of 2003.
2) empirical probability: - derived from observed values
In some cases we can only find the likelihood of a particular outcome by an actual observation.
ex. A baseball player has gotten on base 11 of the last 50 times. What is the probability that he will get on base the
next time.
Note:
Sets: universal set, element, subset, disjoint ==>
Probability: sample space, sample point, event, mutually exclusive
Def. 2.1
A set that contains all possible outcomes for a particular experiment is called the ________________________________
The elements of this set are called _____________
Def. 2.2
Any subset of this set is called an event and the individual elements are called sample points.
Consider the sample space with four sample points: S = { s1, s2, s3, s4 }
E = { s2, s3, s4 }, F = { }, G = { s1, s2, s3, s4 } ==> these are all events, how many different events does S have?
→ __________
The events E1 = { s1 },
E2 = { s2 }, E3 = { s3 }, and E4 = { s4 } are called
elementary events of S ---- events that contain one single sample point
List some of the other events:
examples:
Def. 2.3
If each of the elementary events of S are equally likely to occur, then we say S has _________________________
or S is ______________
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Examples:
W2-1: A die is tossed. What is a sample space ?
What are the sample points ?
S = { ________________ } with sample points ==> ______________________
the elementary events:
the probability of each elementary event if sample space has uniform probability? ______
ex. W2 – 2: A coin is tossed three times. What is a good sample space ? What are the sample points ?
S = { hhh, hht, hth, thh, tth, tht, htt, ttt } are there other sample spaces ?
the elementary events:
the probability of each elementary event if sample space has uniform probability? ______
ex.W2 – 3: A student takes a 5 problem multiple choice question. Each question has three possible choices only one of
which is correct. What is a good sample space ? What are the sample points ?
S = { aaaaa, aaaab, aaaac, aaaba, ... }
How many different sample points are there ( n(S) = ___ ? ) ,
the probability of each elementary event if sample space has uniform probability? ______
Whenever E is an event of some sample space S with uniform probability, we define the
probability of E by
# of elements in event E
P( E ) = --------------------------------------------# of elements in the sample space S
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ex.
consider a toss of a coin - suppose that n represents the number of times a coin is tossed and f1 the number of heads in
the tosses, f2 represent the number of tails ( f1/n and f2/n represent relative frequencies)
Number of trials
frequency of heads
n
f1
frequency of tails
f2
relative freq. of heads
f1/n
relative freq. of tails
f2/n
1
5
10
100
500
1000
if we allow n to go to infinity , relative frequency of each event  ________ and ________
ex. roll a fair six-sided die with each side having the same likelihood of occurring repeat as previous example
as n → ∞, the relative frequencies approach what number ? _______
Note:
A probability model of an experiment ( the probability distribution ) is when we assign a probability to each
of the elementary event so that the properties below are satisfied.
ex. S = { s1, s2, s3, s4, s5, s6 , s7, s8 } , E = { s1, s3, s5 } . If S has uniform probability, then
P ( E ) = ______
the probability distribution of S is given by:
ex. Suppose that we have a six sided die in which the six faces are labeled as 1, 1, 1, 2, 2, 3
Find a sample space , its elementary events , and its probability distribution. Does the sample space have uniform
probability.
ex. Ten cards are labeled 1-10. Do as above example.
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Properties of Probability.
1. P( S ) = __________
2.
P (  ) = __________
3.
_______ P(E ) 
______
4. The sum of the probabilities of all the elementary events is = _________
5. If S has uniform probability , then P (E ) = n(E ) / n(S )
Note: If S does not have uniform probability but it does represent a good probability model, then
1)
P ( E ) =  P( elementary events ), sum of the probabilities of the elementary events
2) and 0  P(E)  1
ex. S = { s1, s2, s3, s4, s5 } with p({s1}) = 1/20, P({s2}) = 3/20, P({s3}) = 7/20, P ({s4}) = 7/20,
then P ( { s5}) must equal ? ________ . Does S have uniform probability ? ____________
If E = { s2, s4, s5 } What does P ( E ) = ? ____________
ex. Find the probability of each of the following
1) A ball is selected at random from a box that contains 6 red, 4 blue, 8 white, and 2 black balls. What is the
probability that the
ball is red ? _________
either red or white ? ___________
What is a good sample space ? ______________________________ Probability Distribution ?
2) A pair of dice are rolled what is the probability that the outcome will be a sum greater than 10 ? __________
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3) A card is selected at random from a standard deck of card :
52 cards – four suits; hearts, diamond, spades, clubs -- 26 red – 26 black,
Find a sample space.____________________________________________ Does it have uniform probability ?
What is the probability that the card selected is an ace ? ___________
a diamond ? ________
not a face card ? _______
4) A company has the following employee classification ( 500 employees ). A person is selected at random.
Find the probability that the person selected is
a) a woman . ________________
b. a man in sales ___________ c. in management ________
d) a woman if the person is known to be in management ? ___________
women
men
management
20
50
sales
100
160
other
150
20
5. A poll is taken from a group of 100 individuals. It is found that 40 smoke cigars, 35 smoke cigarettes, and 50 do
not smoke. A person is selected at random. What is the probability that the person selected
a ) smokes only cigarettes ? ___________
c) smokes at least one type ? __________
b) does not smoke ? _____________
d) smokes cigars if he/she is known to smoke ? ________
21
6. A card is selected at random from a standard deck of cards. What is the probability that the card selected is
a) an ace ? ________
b) a diamond ? _____________
c) a face card ? ________
c) an ace if you are told the card is a diamond ? ________
22
Counting Formula:
The following will help us in counting the possible outcomes of a set. It is not always feasible to
list all the possible outcomes.
Let A = { 1, 2,3 } , B = { 4, 5 } . Find n( A  B ). __________
If A = { 1, 2,3 ,... 10 } and B = { 30, 31, 32, .. 40 }, then n ( A  B) = _________
What if A = { a, b, c } and B = { a, e, i , o, u } ? n( A  B ) = ________ and n (A  B) = __________
Conclusion:
For any two sets A and B, n( A  B ) = n( A ) + n( B) – n ( A  B).
Let A and B be any two subsets of some universal set U. Then n( A  B) = n( A ) + n( B) - n( A  B )
unless A and B are ___________________. In that case n( A  B) = __________________
Also, notice that if n( A ) is known, then
n( A / ) = ___________________
ex. A group of 20 students are enrolled in a math class and a group of 30 students are in an English class.
If the total number of students being looked at consists of only 41 students (each of which must be in at least one of the
two classes). Then how many of these are taking
both Math and English ? _________
ex.
How many are taking math but not English ? _____________
Ten students took a quiz on Monday and 25 students took a test on Wednesday. If 4 students took both
the quiz and the test, then how many students took at least one (of the quiz and the test ) ?
23
ex.
From a group of 100 the following information is obtained. In the last year
60 have driven a car
40 have driven a truck
25 have driven an SUV
5 have driven all three types
10 have driven a car and a truck but not an SUV
10 have driven both a truck and an SUV
***** 20 have driven a car and a second vehicle
How many have driven
a car only ? ___________
none of these ? __________
at least one of the three ? _________
With respect to probability :
1. S: is called a certain event because P(S) = 1 ,
 : is called an impossible event because P (  ) = 0
2. Notice that n(S)  1 but P(S ) = 1. Also, n( ) = 0 .
3. If A and B are disjoint then A n B =  but A  B  0. However, n( A  B ) = 0 but n( A  B)  
4. E: for any event E in the sample space S, 0  P ( E )  1.
5. Elementary events: the sets that contain only the sample points ( single element sets - { s1 } ) are called elementary
events. The actual elements of S are called sample points --- s1, s2, s3, ...
6. If each elementary event is equally likely to occur – sample points have the same probability of occurring – we say the
sample space S has uniform probability
7. A sample space S is said to have a valid probability model if
1) the sum of the probabilities of the elementary events is 1,  P(elementary events ) = 1 , and
2) 0  P( E )  1, for any event E.
3) P ( S ) = 1
4) P(  ) = 0
Additionally,
n(E)
8) P( E) = -------- , if S has uniform probability.
n(S )
9) If E = { s1, s2, s3 }, then P ( E ) = P ( { s1}) + P ({s2}) + P({s3}) - this can be used even if S is not known to
have uniform probability.
24
Odds in favor ( Of ) Odds against ( Og )
We define the odds in favor of an event E occurring as Of = P(E) / P(E/ )
the odds against an event E occurring is defined as the reciprocal of E;
O g = P(E/ ) / P(E)
ex. What are the odds in favor of rolling a double in a roll of a pair of fair dice ?
ex. A person is to be selected at random from a group of 40. If you are in that group, what are the odds against you being
selected ?
ex. If the odds in favor of an event E occurring is 1:3, then what is the probability of that event occurring ? ________
More Examples:
ex. Use the following Venn-Diagram to answer the questions about probability.
Find
a) P ( A ) = ___________
b) P( A  B) = ___________
c) P( A  B) = _______________
d) P ( A /  B/ ) = ________________
e) P ( A  B/ ) = ______________________
25
ex. Use the table that follows to find the given probabilities.
in favor of strict gun
against strict gun
control laws
control laws
no opinion
totals
women
240
60
20
320
men
220
380
80
680
360
440
100
1000
totals
A person is selected at random. What is the probability that the person is
a) a man ? __________________
b) a man that is against strict gun laws ? ___________
c) against strict gun laws if the person is known to be a woman ? ___________________
d) not a (woman that favors strict gun laws) ? _________________
e) a woman or a person that is favor of strict gun control laws ? __________________
Notice that since
n( A  B) = _______________
and n( A / ) = ____________ We have the following ideas with respect to probability.
P( A  B ) = _______________________________
P( A / ) = ___________________
26
Short Quiz – September 22, 2004
Name _____________________________________
examples:
1. A card is drawn at random from a standard deck. What is the probability that the card is an ace or a red card.
2. The local weather forecast is 30 % chance of rain. What is the probability that no rain will fall ? ________
3. A 12-sideded ( called ? ) has the sides labeled as { 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4 , 5 }. The die is rolled.
a) what is a good sample space ? S = ______________________________________
b) Does your sample space have uniform probability ? __________
What is the probability of getting
c). a number greater than 5 ? ______________
d) a prime number ? ___________
e) a prime number or an even number ? ______________
4. There are 60 objects that can be classified as being red objects and there are 40 objects that can be classified as being
heavy objects. Twenty of the red objects are not heavy. An object is selected at
random what is the probability that the object is
a) red or heavy ? _______________
b) neither red nor heavy ?
5. 500 plants are classified as having fruit, deep roots, broad leaves.
250 had fruit, 200 broad leaves, and 100 deep roots
75 had fruit and broad leaves, 50 had broad leaves and deep roots, 40 had fruit and deep roots
20 had all three types of classifications.
One is selected at random. What is the probability that it can be classified in at least one of the three types ? _______
27
Conditional probability –
ex. Construct a Venn diagram to answer the questions that follow.
A survey indicated that out of 200 individuals: 70 read the newspaper early in the morning, 100 watched the news in the
evening. 20 watched the news in the evening but did not read the newspaper.
A person is selected at random.
What is the probability that the person either read the newspaper or watched the news?
______
If the individual is known to have read the newspaper, then what is the probability that he/she also watched the news ?
______
ex. A card is selected at random from a standard deck What is the probability that the card is an ace ? _____________
If you catch a glimpse of it and you notice that it happens to be red, then what is the probability that it is an ace ? _____
ex. A pair of distinct dice ( a red and blue ) is rolled. What is the probability that
a) sum is less than 4 ? _____________
b) the red die is a five ? ____________
c) a double if you know that the red die is a six ? ________________
d) sum is greater than 7 , if you know that the red die is a four ? ___________
Let A and B be any two nonempty events.
We write P(A | B ) or P( A / B ) to mean “the probability of A given that B has occurred.” ( conditional probability)
P( A  B) means “the probability that both A and B occur”
P( A  B) means “ the probability that either A , B, or both occur”
28
Def. We define P(A | B ) as
P( A  B)
P(A | B ) = --------------P( B )
This represents the probability that the first set (A in this case) occurs given that the second set (B in this case) is known to
have occurred
We can write each of the following probabilities regardless of the relationship between A and B.
1)
P ( A  B ) = P(A) + P(B) – P(A  B )
2) P ( A / ) = 1 – P( A ) , and
*** 3) P( A  B ) = P( A )  P( A | B ),
Notice where this last property comes from:
From the definition of P( B | A ) we have P( B | A ) = P( A ∩ B ) / P ( A ). Using some algebra, rewrite
P( A ∩ B ) = P( A ) P( B | A )
In the case that A and B are independent events ( the fact that A has occurred does not change the prob. that B will
occur ) ,
we reduce
P ( A ∩ B ) = P ( A ) P ( B ) ---- we discuss this idea in a little bit.
ex. If P( A) = 0.4, and P( B) = 0.8, can A and B be mutually exclusive ? Why or Why not ? _____
Using the probabilities above assume that P( A  B ) = 0.9. Find
P( A | B ) = ______________
what about P( B | A ) = _____________
ex. If P( A ) = 0.6 , P( B ) = 0.5 , and P( A  B ) = 0.3, then find
a) P( A  B ) = _______________
b) P ( A | B ) = _______________
c) P( B | A ) = _______________
There is a special case in which P(A | B ) = P(A ) , that is , it does not matter that B has occurred - the probability of A is still
the same as before.
29
Def. (INDEPENDENT).
Let A and B be any two given sets. We say that A and B are independent provided
Either P(A | B) = P(A) or P(B | A ) = P(B).
Note that this means:
Knowing additional information did not change the probability
ex. A class consists of 50 students; 20 male and 30 female. There are 25 students with long hair, 10 of them are male.
A student is selected at random. What is the probability that the student
a) is male → ___________
b) the student has long hair → _____________
c) Is being male and having long hair mutually exclusive events ? ________________
d) If a student is known to have long hair, what is the probability that the student is a male student ? ___________
e) Is being male and having long hair independent events ? _______________
While
every group of disjoint events happen to be → mutually exclusive
every pair of independent events happen to be → intersecting
every pair of event sets that are intersecting do not have to → independent
It does not mean that if two events intersect, then they must be independent ( this statement is false )
ex.
Given the following Venn Diagram determine the probabilities. Determine if E and F are independent or not.
P(A ) = 0. 4
P(B ) = 0.2
P( B ∩ A ) = 0.08
a) Are they independent ? _______________
b) P ( A | B ) = _______
c) P( B | C ) = _______
30
ex.
30 cards are labeled from 1 – 10 ( three of each kind ) one is red, one is black and one blue.
a) A card is selected at random and then replaced. A second card is drawn . Continue.
What is the probability that three ( three cards are drawn ) cards are black ? ___________________
What is the probability that none of the cards are black ? __________________
At least one of the cards is black ? ___________________
ex. Suppose that A and B were mutually exclusive events with neither being nonempty.
Find P( A | B ) = __________
ex. Suppose that A and B were independent events. find
a) P( A | B ) = ___________
b) P( A  B ) = _______
ex. Four lights are independent of each other. What is the probability that a person will
( Assume that the probability that a light is green is 3/5 )
a) get all four green lights
____________
b) get the first light to be green
_____________
c) will get at least one green light
_________________
It is useful to use a tree diagram to answer questions about conditional probability.
The following example illustrates what each part of the branches represents.
31
ex. Two balls are drawn from an urn that contains 4 red, 7 blue, and 9 white balls. If the second ball is drawn without
replacement find the probability that
both balls are red ?
___________
the first is red and the second one is white ? _____________
the second one is white if you know that the first one is red ? _________________
the second one is white ? _______________
the first one is red if you know that the second one was white ? ____
Use the following tree diagram to answer the questions that follow.
a) find P ( A / ) = ________
b) P ( D | A ) = _______________
c) P ( D ) = ____________________________________________________________________________
d) P ( A | D ) = ________________________________________________________________________
32
Bayes’s Formula:
A sample of 20 students in a math 1312 class is taken. On any given day 15 out of 20 are on time to class.
If a student is known to be on time, the probability that they passes exam one is 0. 8.
If the student is known to not be on time, the probability that they will pass an exam is 0. 4.
A student is selected at random.
Find the probability that
a) the student is not on time. _____________
b) the student did not pass the exam if he/she is known to be on time. __________________
c) the student passed the exam . ___________________
d) the student was on time if he/she is known to have passed the exam. ________________
Birthday Problem: What is the probability that
Two people (out of two ) will both have a birthday on July 15 ? __________
Two people will have the same birthday. ___________
(on the same day – any day )
Two people will not have the same birthday ________________
None will have the same birthday on the same day ? (out of 2 ) ____________
Exactly two out of three will have matching birthday ==> __________
No two out of five will have matching birthdays ? ____________________
(In a class of 35) What is the probability that at least two will have the same birthday ? __________________
33
Factorials
The process of multiplying numbers of the form
1 • 2• 3
1 •2 • 3• 4 •5 •6 •7
1 • 2 • 3 • 4• ... • 100
occurs frequent enough that we define this type of multiplication.
Def. We define 0 factorial, 0 ! , by 0 ! = 1.
Let n be any natural number, say 1, 2, 3, 4, ....
We define n factorial as
n ! = n ( n – 1 ) ( n- 2 ) .... • 3 • 2 • 1.
Look at these examples to better understand the meaning of n !.
4 ! = 4 • 3 • 2 • 1 = 24 ,
2! =2•1=2
10 ! = 10 • 9 • 8 • 7 • 6 • 5 • 4 •3 • 2 • 1,
1!=1
200! = 200 • 199 • 198 • ... • 3 • 2 • 1
Other examples:
8!
= __________
10!
100!
= ____________
98!
Notice:
1) 1 ! = 0 ! = 1
2 ) n ! gets big very quickly → 5 ! = 120 , 6 ! = 720 , 7 ! = ......
More Examples:
1.
7 ! - 6 ! = ____________
2.
3. 5 ! + 3 ! = _____________
5.
4. 20 ! / 19 ! = ___________
4!
= _________
1!0!
ex. 6 ! = ___________
4 ! - 0 ! = ___________
6.
8 ! / 6 ! = _____________
6!
= _____
3!
4 ( 3 ! + 2 ! ) = __________
Other Factorial Problems
( n ! ) • (n + 1 ) = _________
5!
--------- = _________
4! -5!
(n–1)!
/ ( n ) ! = _______________
80! = _________
300 ! / 298 ! = _________
34
Notation and a brief introduction to permutations and combinations.
Def. A permutation is an arrangement of the objects of a given set with the following conditions.
a) no repetitions and
b) order matters.
ex. { a, e, i, o, u }
Find every possible grouping of four of the five vowels in which order matters ( four letter “words”)
ex. Given six books – four are to be selected and presented to the class
How many different book presentations - order matters – are possible ?
ex. A club with 12 members is to select a President, a Vice President, and a Treasurer. How many different seletions are
possible ?
Def. A combination is an arrangement of objects of a given set with the following conditions
a) no repetitions and
b) order does not matter.
ex. { a, e, i, o, u }
Find every possible way that you can select four of the five vowels in which order does not matter . __________
ex. A library offers six books over a topic that you are interested in – you can only select four of the six at a time.
How many different books selections are possible ?
ex. A club with 12 members is to select a committee of three. Any three can serve in this committee. How many different
committees are possible if there is no distinction between any of the three members chosen ?
How do we answer each of these six questions ?
We will get back and work these problems in detail – for now we just want the formulas that we will use.
P(n,r) =nPr =
n!
(n  r )!
and
nCr
= C(n,r) =
 n
n!
  =
 r  r!(n  r )!
For example:
P( 6, 2) = _________
P(50, 3) = __________
C( 40, 2) = _______
C( 20, 3) = _________
35
Introduction
Permutations and Combinations
There are problems in mathematics which involve counting, such as how many groups of 5 could you make from
a group of 20 ? That’s easy enough → 4 .
But what if we ask the same question in a slightly different form.
A class consists of 20 students.
A group of 5 students is to be selected from the class. If only one group is selected, how many different ways could that
group be selected ? ABCDE is the same group as ABDEC but not the same as ABDEF.
This is called a combination. The question really is “how many ways can a group of 5 be selected from a group of 20”
or even better → “How many distinct combinations of five people can be made from a group of 20 – keeping in mind
that on any instance only one group of five is being selected.”
If we label the students as S = { s1, s2, s3, ... s20 } we want groups that look like
s1, s2, s3, s4, s5
s1, s2,s3,s4,s6, ...., s2,s4,s6,s8,s10,
s7,s8,s1, s20,s17, and so forth
In the problem above – order did not matter – but what if order did matter.
If we label the students as S = { s1, s2, s3, ... s20 } we want groups that look like
s1, s2, s3, s4, s5
s1, s2, s3, s5, s4
s1, s2,s3,s4,s6, ...., s2,s4,s6,s8,s10, s7,s8,s1, s20,s17, and so forth
keep in mind that order does matter and we have no repetitions - Now how many groups are possible ?
ex.
Five students are to be selected from a group of 20. The first student selected will get $20, the second $10,
the third $5, the fourth gets $1, the fifth gets $0.50. Order obviously matters! Is this the same question as
the one above ?
This is an example of a permutation. The question remains almost the same – except order matters !
“ How many ways can a group of five be selected from a group of 20 – if order matters”
or even better “How many distinct permutations of five can be made from a group of 20 – keeping in mind
that you are only using one group of five at any given time”
This problem does have a solution:
we are making five decisions: 1st person, 2nd, 3rd, 4th, and 5th person
1st: there are 20 ways of making this decision, 2 nd: there are 19 ways of making this decision, ...
( What does this sound like ?
The multiplication principle says: multiply these numbers:
20 • 19• 18•17 •16 = __________
In either case above, we implied that no person was to be selected more than once and there were to be no repetitions.
36
Combinations: order does not matter ( AB = BA ) and there are no repetitions
Permutations: order does matter ( AB ≠ BA ) and there are no repetitions
More examples. (identify as a permutation or a combination) – if the problems is a permutation – go ahead and solve.
1. How many ways can a Pres., Vice P be chosen from a group of 12 _________
2. Two new federal judges will be appointed from a group of 20 state judges. How many different selections
are possible ? If 12 are men and 8 are women , what is the probability that they are both women ?
___________________
3. How many ways can a committee of 4 be appointed from a group of 8 men and 4 women ? ____
What is the probability that all four are men ? ___________
What is the probability that exactly two are women ? __________
What is the probability that at least one is a man ? ____________
4. How many ways can 6 books be arranged on a shelf ? __________
5. How many different “ 4-letter words” can be made using the blocks ABCD ? ________
how many “two-letter words” if any two of the blocks can be used ? _______
6. Consider all five card draws from a standard deck of cards. How many different hands are possible ?
What is the probability that all are hearts ? ______________
What is the probability that at least one is a heart ? ____________
What is the probability that exactly three are hearts ? __________
Permutations: no repetitions, order matters, ABDC is different than ACBD Combinations: no repetitions, order does not matter ABDC is the same as ACBD
To actually count the number of permutations we begin by introducing the multiplication principle and then use to construct
formulas for permutations and combinations.
37
Multiplication Principle of Counting – Fundamental Principle of Counting
Cross product A x B ( Cartesian Product )
If A = { a, b, c } and B = { 1, 2, 3, 4 }, we defined A x B by saying that it consisted of pairs of objects constructed by
matching all the elements in A with all the elements in B
A x B = { ( a, b ) | a  A and b  B }
In this example: A x B = { (a,1), (a, 2), (a,3), (a, 4) , (b,1), (b,2), (b, 3) .... (c, 2), (c,3), (c,4) }
Having remembered the cross product of two sets ( or more as discussed earlier ) . We bring up the MPC or FPC
Multiplication Principle
This method allows you to count all the different of making several decisions at once.
If a decision can be made in r ways and a second decision can be made in s ways, then both decisions can be made in
rs ways(multiply.
ex. You wake up in the morning and decide what to wear. You have a choice of any one of three pants and any one of
five t-shirts.
How many outfits are possible ? You have two decisions to make:
pants ( there are r = 3 ways of making that decision )
t-shirt ( there are s = 5 ways of making that decision )
According to the MP, you can make both decisions in r • s = 3 • 5 = 15 ways. So, there are 15 different outfits.
ex. Second Example.
Now have three pants, five shirts, and four pairs of shoes – How many different outfits ? ____________
ex. A board consists of four members - one represents management ( 5 available to serve),
accounting dept ( 4 available) , sales dept ( 10), and one from the staff classification (20). How many
different boards could be created – in terms of who is currently serving ?
ex. How many different menus are possible if there are four salads, three meats , and six desserts
and the diner must eat only one of each type ?
___________
ex. how many different codes are possible if the code consists of sex, two digit age, and one letter of the
alphabet ( case sensitive )
Fundamental Principal of Counting:
Decisions D1, D2, ... can be done together in r1r2r3 .... ways
38
Permutations and Combinations
Problems and Formulas
Permutations:
1) no repetitions and 2) order matters
ex. { a, b, c , d }  some three letter permutations 
ex. How many ways can a Chairman and a Vice Chairman of the Board be selected if they are to come from a
group of a 10-member board ?
ex. Five men walk into a bar with 12 stools. How many distinct seating arrangements are possible ?
ex. Three men and three women come to a party and sit on a long bench. How many seating arrangements are
possible ? ______________ How many if they are to alternate (male-female-male-... ) ? __________
Formula: P( n, r ) =
n
P
r
=
39
Combinations:
1) no repetitions and 2) order does not matter
ex. { a, b, c, d }  some three letter combinations 
ex. How many ways can a two member committee be chosen from a 10-member board ?
ex. An A&M student goes to the library and checks out three out of the five books available. All five books are
different(distinct). How many different groups of three are possible ?
Formula: C(n,r ) = nCr =
 n
  =
r
40
Other examples:
1) 34/ 546 : Eight horses are entered in a race. In how many ways can the horses finish ?
What is the probability that the top three favored horses finish in the order 1,2,3 ?
2) 41/ 546 How many ways can a 10 –question multiple be answered if each question has four different answers ?
What is the probability that the first and last questions are correct ? (all other questions are wrong)
3) 48/ 547
A poker hand consists of 5 cards dealt from a deck of 52 cards. How many different poker hands are possible ?
How many with four aces ? _____________
What is the probability that you have four aces ? ____________
4) In how many ways can a hand consisting of 6 spades, 4 hearts, 2 clubs, and 1 diamond be selected from a deck of 52
cards ? What is the probability that that happens with a 13-card hand ?
5) How many different sequences are possible in 4 rolls of a fair six-sided die ? _____________________
What is the probability that the sequence consists of all sixes ? ____________
What is the probability that the sequence consists of no sixes ? _______________
41
4/551: Two men and a woman are lined up to have their picture taken. If they are arranged at random, what is the probability
that
a) the woman will be on the left in the picture ?
b) the woman will be in the middle of the picture ?
8/551 Keys for older General Motors cars had six parts, with three patterns each.
a) How many different key designs are possible for these cars ?
b) If you find an older GM key and you own an older GM car, what is the probability that it will fit your trunk ?
14/551 10 –questions with 15 possible matches, no repetitions – what is the probability of guessing and getting every answer
correct ?
20/552 12 girls at random from a freshman class : 200 freshman girls, including 20 from minorities, and the principal would
like at least one minority girl to have this honor. If he selects the girls at random, what is the probability that
a) he will select exactly one minority girl ?
b) he will select no minority girls ?
c) he will select at least one minority girl?
42
32/553
43
Brief Review of Matrix Operations
Matrices are rectangular array of numbers that describe objects in given sets.
ex. A company produces red and blue bicycles ( Each bike-is of one color). They are either
traditional bikes or mountain bikes.
Here are the numbers of bikes produced on Monday;
200 red mountain bikes, 100 blue mountain bikes, 150 red traditional bikes, and 50 blue
traditional bikes.
Mountain bikes sell for $150 and traditional bikes sell for $100.
Write a matrix that describes the above situation as well as the amount of revenue for the bikes produced on
the given day.
Let M = Mountain Bikes and T = Traditional Bikes
If you are interested in describing the amount of revenue generated by the bikes in terms of their color →
M
cost matrix C =
150
T
100
blue red
200 100
Quantity Matrix Q = 

150 50 
mountain
traditional
the product of both of these matrix will tell us the amount of money that that is generated by red and blue bicycles.
C•Q
If we wanted to know the value generated by mountain –vs- traditional bikes we would write Q as follows
M T
200 150
Quantity Matrix Q = 

100 50 
blue
red
C•Q
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General Notation:
of matrices
1) rectangular array of numbers with rows and columns
We normally use capital letters to name the matrices.
A=
3
,
B=
3
1 / 5,
C=
2
3
4
5
1
6
7
8
9
0 
D= 
, E=
  1 10  2  3  4


 5  6  7  8  9 
3  2
5 7  ,


4
7 
 
2) Dimension of a matrix: m x n
We use the number of rows and columns to describe the matrix.
A is a ___________ matrix
C: __________
B is a ____________ matrix
D: _____________
E: ____________
3) elements of a matrix: aij
Look at matrix C: we can label the elements of C as follows:
Look at matrix E: we can label the elements of E as follows:
Look at matrix D: find each of the following entries (elements)
d13 = _________
d32 = _________
d42 = __________
d25 =__________
Special Types of Matrices:
Zero Matrices: All entries are zero
2x2 zero matrix
1x5 zero matrix
4x3 zero matrix
Square Matrix: A matrix that has the same number of rows as columns
1 2
A = 5 , B = 
 C=
3 4
1 0 0
0 1 0 


0 0 1
45
Identity Matrices: diagonal entries – a11, a22, a33,.... are all = 1 while all other entries = 0
1
1 0 0 
0
1 0
1 ,   , 0 1 0 , 
0
0 1  
0 0 1 0

0 0 0
1 0 0
, ....
0 1 0

0 0 1
Addition: add corresponding entries so that you end up with a matrix that resembles the original two in size- this can
only occur if the original matrices are identical in size .
A + B is defined if A : m x n matrix, then B must also be m x n matrix.
6
3  2 +   = _________
 2
1  2 4  3  ___
2  1 + 0  3 =  ___

 
 
0 4  1 2   ___
___ 
___ 
___ 
Subtraction: if treat matrices as real numbers, we can use addition.
Let - A represent the opposite of matrix A. Then B – A = B + ( -A).
2  3  3  2
 ___
4 0  -  1 1  =  ___

 


___ 
___ 
 4   2 
 ___ 
  2 -   2 =  ___ 
   


 0   3 
 ___ 
There are two types of products of matrices –
multiplication by a scalar (nonmatrix – real number)
multiplication of two matrices
Scalar Multiplication: easy product - distributive law
 3   ___ 
a) 4   = 

  2  ___ 
 2 1  ___
c) - 2 
 = 
 2 1  ___
b) - 2 2  3 1 0 = ___ ___ `___
___ 
___ 
___ 
46
Some Simple products of Two matrices:
If we multiply matrix A by B( in that order), then the number of columns of A must be the same
as the number of rows of A. If A is an m x p matrix, then B must be a p x n matrix
ex.
2
 1 • 1  2 = ?
ex.
4
1  2 3 •  0  =
 3
 1  2  1
  2  • 1 0  = ?
  

ex.
4
1 • 2  3 =
 
In the two examples above, what do you get if you change the order of the matrices ?
 1 2 
ex. 1 2 3 •  0  2 =


 4
1 
General Product of Matrices
1 2
 1  2 3
ex. 
•

  2 2 4 =
3 4


2  3
 1  2 3


ex. 
 • 2 3  =

2
2
4


4 1 
47
Markov – Chains –
Transition Matrix: has probabilities (transition probabilities)
they describe the probability in moving from one state to another.
Credit Card: Let C = event credit card is used.
Previous experience tells a company that
P( C next month | C prev. month ) = 0.8
P( C next month | C/ prev. month ) = 0.3
P( C/ next month | C prev. month ) = 0.2
P( C/ next month | C / prev. month ) = 0.7
This can be illustrated with a tree diagram as well as with a matrix.
Transition
matrix
Given
Month
next month
uses card
card is not used
card is used
card in not used
0.8 0.2
0.3 0.7


Initial –probability vector
Prob. woman uses card = 0.9
[ 0.9
prob. woman does not use card: _____
0. 1 ]
What are the probabilities for the second month ? ____________
Look at example on page 558:
Initial Probabilities
[ 0.6
0.3
0. 1 ]
Transition Matrix.
0.2 0.6 0.2
 0.1 0.5 0.4


 0.1 0.1 0.8
What are the probabilities after one generation ?
48
Name _____________________________ Math 1312.010 – May 29, 2002 – quiz
1.
{ 0, 1, 2, 3, 4, 5, ... } is called the set of ___________________________
A real number that can be written as a fraction is called a ______________________________
2. True or False.
_______________ a. Every real number is either positive or negative.
________________ b. All odd natural(counting) numbers are prime
________________ c.
3.
1 is the smallest natural / counting number
Which of these are well-defined sets ? State Why or Why not ?
a. All students enrolled in math 1312.030 at ASU during this fall 2001 session that are here today.
b. All students in this class that drove fast to school.
4. If a set does not contain any elements we call it the ______________________________
A set that contains all objects under consideration is called the ________________________________
5. Let A = { x : x is even } and B = { x is prime }
, U = { 2, 3, 4, 5 } where U is the universal set .
a. Find A = { __________________________ }
b. Is 4  A ? _______________
c.
Is 6  A ? ______________
d. Is { 4 }  A ? ___________________
e. Is   B ? ______________________
7. True or False. For any set A
___________ a. A  A
_______________ b. { a, b, b } = { b, a }
____________ { 2, 1} { 1, 2 }
49
Name ___________________________ Math 1312 – QZ #2 – September 4, 2001
1. Let U = { x | x < 10 and a whole number } and A = { x | x 2 = 4 }, B = { x | x is a natural number less than 5 }
C = { x | x is an even number , with x2 > 9 }, D = { x | x is an odd number }, E = { 1, 5 }
a) Find A = { ______________________ }
b) Find D = { _________________________ }
c) Find C  D = { __________________________________________ }
d) B x E = { ____________________________________ }
e) C  E = { ___________________________________________ }
2. Shade the set on the given Venn Diagram
a) A  B
b) A/
c) A n B /
3. The set that contains no element at all is called the _______________________________
4. How many subsets does the set { a, b, c, d, e } have ? _______________
5. True or False.
_________________ a) ( A/ )/
= 
_________________ b) n( B ) represents the number of objects (elements ) in the set B
_________________ c) U/ = 
_________________ d) A  A/ = U
50
Name _____________________ Math 1312 – Qz #4, September 11, 2001
1. If two events A and B are disjoint, then A  B = ___________
2. Complete the formula for any two sets A and B. n ( A u B ) = __________________________________
3. If S = { s1, s2, s3, s4 }, then list all of the elementary events
E1 = { s1 } , ______________________________________________
Find P (  ) = ________________
4. Find P ( S ) = ____________
5. For any event E,
____________  P( E )  __________
6. Suppose that S has uniform probability with S = { s1, s2, s3, …., s20 }, E = { s8 }, and F { s1, s2, s3 }.
Find
a) P ( E ) = _______________
P ( E U F ) = ______________
c) P ( F / ) = ____________
7. If there is a 20 % chance of rain falling today, what is the probability that no rain will fall today ? ___________
8. Given the following table . Find each of the following probability
red
not red
woman
200
120
man
40
140
A person is selected at random. What is the
a) probability person selected is a woman wearing red ? ____________________
b) probability that the person selected is wearing red and is a woman ? ____________________
c) probability that the person is a woman if she is known to be wearing red ? _____________
51
Name ___________________________________ Math 1312 - QZ #4 - September 11, 2001
1. Properties of Probability
Fill in the blank
a) Another word for mutually exclusive events is __________________________________
b) The probability of any event E is always bounded by __________  P(E)  __________
c) P(A ) + P ( A / ) = ______________
d) P( elementary events ) = __________
2. Of 100 students 24 can speak French, 18 can speak German , and 8 can speak both French and German. If a student is
picked at random, what is the probability that he or she can speak French or German ?
3. 200 cars enter an intersection. 40 turn left, 100 go straight, and the remaining cars turn right. One of these cars is chosen
at random. What is the probability that the car will turn ?
4. A loaded four sided ( a _______________ ) die is rolled. The following probability distribution describes the outcomes of
the die
P( s1) = 1/10, P(s2) = 2/10, P(s3) = 3/ 10, P(s4) = 4/10
What is the probability that an even number comes up ? ______________
What is the probability that an even or prime number comes up ? ________
5. A card is drawn at random from a standard deck. What is the probability that the card selected is either a king or a
diamond ? _____________
52
Name ____________________________________ Math 1312 – Qz #2, January 22, 2001
HW #3 page
1. Let U = { 1, 2, 3, 4, 5, } with A = set of all even numbers, B = set of all prime numbers, and C = { x : x > 2 }
a.
Find A u B = { _______________________________ }
b.
Find A n B = { ______________________________ }
c. Find the complement of C, C / = { _________________________ }
d. A - B is defined as the set containing all elements of A that are not in B. Find A - B
2.
Shade the Venn diagram that illustrates each of the following sets.
a.
Two disjoint sets
c. The intersection of A and B
b. The union of sets A and B
d. The complement of A
3.
If A and B are disjoint what does A n B = ? ___________
4.
How many subsets does the set { a, b, c, d } have ? __________
5.
Out of 40 students that came to class today 30 listened to the radio and 20 read the newspaper. Explain these numbers.
53
Name ________________________________ Math 1312 – January 28, 2002 – Short
1.
Quiz (10 points )
The set that contains all possible outcomes of an experiment is called a __________ ___________ (2 words)
2. Any subset of the set that contains all possible outcomes is called an _________________
3.
List all of the sample points of S = { a, b, c }
How many different events does the set S have ? __________
4.
A six sided die is rolled. What is the probability that a four comes up ? ________________
5. A student is to be selected from a group of 20. There are 8 males and 12 female students.
What is the probability that you will be the one selected if the selection is made at random ? _____________
What is the probability that a female student will be selected ? ______________
6.
Six numbers are equally weighted so that they add up to 1. What is the value of each number ? ___________
7.
Let S = { a, b, c }.
True or False
___________ a) { a, b }  S
___________ b ) c  S
____________ d) { a } is an elementary event
Homework: page 505: 43, 44, 45, 47, 49, 52, 53, 54, 55, 56, 57, 58
page 505 7, 10, 13, 18, 19, 22, 27, 28, 29, 30, 35, 37, 38, 41, 42, 60, 61
page 515: 1, 4, 7, 10, 13, 16, 19, 22. 25, 28 page 526 - 1, 3, 5
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