Download Measures of Central Tendency

Introduction
Measures of Central Tendency, Dispersion, Sets and Definition, Counting
Measures of Central Tendency
(and normal distribution)
_
Average ( x )
Arithmetic mean, Expected value
_
Def. Given values x1, x2, x3 we define the arithmetic mean (average), x , by
_
x1  x2  x3  ...  xn
x1  x 2  x3
or in general x =
=
3
n
_
x =
Here, the symbol

x
i
n
represents the idea of adding up all values (sum of all xi )
For example:
1.
During the last five years an employee at a company has received the following pay raises with respect to previous
year’s salary;
2 %, 3 %, 2 %, 0%, and 3 %.
In terms of percents, what is the average raise that this employee has received during the last five years.
---- ( in terms of money: is the answer representative the increase ? )
2. In order to keep costs down
a company charges $5 for shipping within the city, $10 within the state but outside of the city, and $25 for out of
state shipping.
During the last week 100 orders were shipped;
35 were within the city, 40 were in state but outside the city, and the rest were out of state.
What was the average amount charged during the week ? ____________
3.
A student’s grade will be determined by exam grades ( each exam counts twice and there are three exams), HW
average (counts once ) , final exam ( counts three times ).
Find the average if the student has the following grades. _________________
Exam: 72, 85, 82
HW: 90
Final Exam: 85
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4.
A student takes four exams during the semester and a final exam. No other grades will be used to calculate the
end of semester grade.
His four exams were; 80, 74, 90, 60. If the final exam counts twice as much as a regular exam, what score does
he have to make on the final exam to have an average of at least an 80 ?
__________
5. A student is currently enrolled in a class in which he has a 20 % chance of getting an A, a 40 % of getting a B,
a 25 % chance of getting a C, and a 10 % of getting a D. What grade do you expect him to get ? Explain your answer.
6.
You want to find the average height of a person in this classroom.
Process:
7. You want to find the average height of a college student at Angelo State University.
Process:
Population –vs- Sample
population: you have everybody ( all data is available in usable form) that will be used to calculate values. (example 6)
sample: only a small portion of the data is available and you will use it to find ( “estimate” ) values for the entire population.
(example 7 and example 8)
ex 8. You are asked to determine the # of a type C organisms in a pond.
Process:
ex 9. A test is given to 30 students – the class average is calculated. Is this a population or sample mean ?
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A recent graduate has two options to choose from. They both advertise an average salary of
$35,000.
a) average of $40, 000: small company with salaries
b) average of $35,000:
20 K, 20K, 20K, 20K, 20K, 20K , 35K, 35K, 95K, 115K
15K, 15K, 30K, 30K, 35K, 35K, 40K, 40K, 55K, 55K
Which one would be the best fit for you and why ?
We talked about one of several measures of central tendency.
1. Arithmetic mean
_
Given values x1, x2, x3 we define the arithmetic average, x , by
_
x =
_
x1  x2  x3  ...  xn
x1  x 2  x3
or in general x =
=
3
n
x
i
n
In the event that the data is large and can be grouped together in different classes, then we can use the following formula.
_
individually listed
x
=
x1  x2  x3  x4  ....xn
n
given frequencies we can write as
_
x =
x1 f1  x2 f 2  x3 f 3  ...  xk f k
=
n
x
i
fi
n
f1 represents the # of times that value x1 occurs, f2 represents the # of times that x2 occurs,... n represents the total
number of data values ( sum of the frequencies)
ex. A large class of 100 students has met for 5 five days. Here is a description of the number of times a student has been
absent. What is the average number of days that a student has missed.
0 absences → 24 students
1 absence →
32 students
2 absences → 35 students
3 absences →
7 students
4 absences →
2 students
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MODE
value (or expression) that appears with the largest frequency (response that is given the most number of times)
Examples:
1.
30 people are asked for their favorite color;
20 said blue, 8 said red, 1 said black, and 1 said yellow
What was the modal response ? _______
2.
The number of times that a driver has been pulled over during the last five years
0, 1, 0, 3, 0, 2, 1, 1, 2, 1, 1
3.
A class of 40 students is asked the number of times they ate out during a five day period.
1 → 0 times
20 → 1 time
15 → 2 times
3 → 3 times
1 → 4 times
NOTE: The mode must be one of the given values – the arithmetic mean does not.
MEDIAN
the middle value of a given set of data. If no middle value exists, then we choose the average of the two middle
values.
1) . Salary within a six-member department in terms of thousands of dollars
20, 40, 30, 20, 30, 15
2) The number of miles that a person walks per month during a 7 month period.
100, 40, 200, 200, 120, 80, 200,
3) the median value of a home in San Angelo is said to be $67,000
if there were 36,000 homes in San Angelo then explain the median value
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Range:
The difference of the largest and smallest value.
1) Ten people weighed themselves one week after starting a weight loss program.
The following values indicate the amount lost in terms of lbs.
½, 2, 1, ½ , 3, 1, 1, 2, 3 ¼, - ½
What is the range of this data ? → _____________
2) A person is dealt a 5-card hand. The player counts how many diamonds are in his hand.
If there are five players sitting in, then what is the smallest and largest possible range ?
give me an example of the smallest range: ________________________
give me an example of the largest possible range: _________________________
Additional Examples:
A class of 8 eight students are asked the number of times that they ate out during this past week.
Here are their responses:
0, 1, 0, 3, 1, 1, 2, 4, 3, 5
What is the arithmetic mean ?
What is the mode ?
What is the median ?
What is the range ?
A quiz is given to five students. The grades were all identical; 85, 85, 85, 85, 85
What was the arithmetic mean ? __________
Think of the distance of each value from the calculated mean:
What about the average distance from the mean ? _______________
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Define another term.
Find an average of the distances from the arithmetic mean – average deviation.
Examples:
A company believes that on the average a bottle of pills contains 50 pills.
Five bottles are selected at random and the pills are counted; 50, 48, 50, 50, 52
Is the average 50 ? _______
Describe the deviation of each bottle ( # of pills in each bottle) and then find an average of this deviation.
Problem?
A second example:
Three questions are given in class to a group of five students. The data below represents the number of problems
missed.
1, 0, 0, 3, 1
Find the arithmetic mean : __________________________
Find the average deviation : ___________
Problem ?
Instead of finding the average of the deviation why not find the average of the squared-deviation.
Example 1:
Example 2:
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Average Squared Deviation:
Asd =
 (x
_
i
 x) 2
n
and if the data is in terms of frequencies →
 (x
_
i
 x) 2 f i
n
This will eliminate the problem we had.
This gives two more ways to look at the distribution of data values.
Variance:
 (x
_
i
 x) 2 f i
Standard deviation:
n
var iance
We use the formulas above if the entire population is known.
In most cases we do not know the entire population – so we use sample variance and sample standard deviation.
Sample Variance:
 (x
_
i
 x) 2 f i
n 1
sample standard deviation = s =
sample _ var iance
Another Example
A class of ten students meets five times per week. The following represents the number of times that each student
attended during the week. Find the arithmetic mean and the average of the deviations.
0
1
0
3
1
1
2
4
3
5
7
A study is done to determine the number of accidents that a student has been involved in.
A sample of 50 students is done with the results that follow
6 have been in zero accidents
29 have been in 1 accident
12 have been in 2 accidents
3 have been in 3 accidents
None have been in more than 3.
Find the sample standard deviation.
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Histograms are graphs of data in which rectangles are used to represent the frequency of each value ( later: the probability )
The rectangles are of width one unit, centered at each value.
ex. 1, 1, 1, 2, 2, 3, 3, 3, 3, 3
f
54321x
1
2
3
Use the following histograms to find which data has the largest mean and which one appears to have the largest standard
deviation.
a)
b)
c)
Can you find the standard deviation of each group of data values.
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Discrete Data –vs- Continuous Data
Discrete Data:
You meet five traffic lights – how many were red ?
Ten employees were hired five years ago. How many are still with the company ?
A company serviced 20,000 accounts this year . How many were from female customers ?
Continuous Data:
A company plans to test to see how much sugar has to be added to lemonade before a customer is satisfied.
Provide exact amounts that will please each of five customers. Assume there is a minimum of zero sugar added and a
maximum of 5 teaspoons.
A sleep depravation test will measure the exact amount of time that five individuals will be able to stay awake.
List the possible amounts. Assume that every individual managed to stay awake at least 10 hours and at most 48 hours.
A room has 10 lights. The exact lifespan of each is found ( when the light ceases to work). Write down each possible
lifespan.
Normal distribution corresponds to a type of data that we can treat as being continuous data – although not entirely true.
When data has a normal distributionwe get the following curve to represent it; a normal curve.
______________________________________
A normal curve has a high point – and it occurs at the mean µ. The area under the curve will add up to 1 square unit.
We know a few curves whose areas are easily found – this is not one of them.
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Normal Distributions
Normal Curves - - mean (µ ) , standard deviation ( ) , inflection points, area under a curve,
_______________________________________
Standard normal curve
If the normal curve has  = 0 and a variance = standard deviation = 1, we call it a standard normal curve.
Do the following values represent standard normal curves ? Why or why not
a) µ = 0,  = 2
b) µ = 0 ,  = - 3 ( ? )
c) µ = - 1 ,  = 1
Do they represent normal curves ? ____________
How many normal curves could you create ? _________
We use tables to find area under a curve. Notice that half of the area is to the right of the mean, half to the left (symmetric ).
We have a function that expresses the curve and there are ways of finding the area under a curve.
f(x) =
1
 2
( x )2
e
2 2
ex. f(x) = 4 . Find the area under the curve between x = –2 and 2
ex. f(x) = 2x. Find the area under the curve between x = 0 and 4
ex. f(x) = x2. Find the area under the curve between x = -1 and x = 2
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It is not as easy to find the area under a normal curve.
Consider the following functions: (see page 610)
f(x) =
1
 2
( x )2
e
2 2
This is the function that we would try to work with when finding the area under a
a normal curve. You can see the problem that we would have.
A table is constructed for a standard normal curve by using techniques that are for the moment out of reach.
If we are given a table with values that represent areas under a standard normal curve
we use the following formula and this table to find areas under a normal curve.
z=
x

Table
z
0
1
2
3
4 ….
9
------------------------------------------------------------------------------------------------------------------------:
:
:
2.1
ex. Find the area to the left of - 2.00 under a standard normal curve.
ex. Find the area to the left of 12 under a normal curve with  = 20 and variance = 16.
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ex. Find the area to the right of 190 under a normal curve with  = 200 and variance = 81.
ex. Find the area between 20 and 30 under a normal curve with mean = 28 and variance = 25
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Introduction to Sets
Def 1.1: 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 at the end of current semester
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 – or choose not to list them:
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,... }
We can use what’s called set-builder notation.
This way we do not write every member of a set –
just indicate what the members look like(what property they have in common).
{ 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 student’s 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 }
Def 1.2: Elements
The members of the set are called elements of the set.
member of a set.
We normally use capital letters to represent sets
We use the symbol  to express the fact that an object is a
and lower case letters to represent elements of a set
A = { a, b, c } or B = { 5, 10, 15 ..... } , C = { x | x is a positive real number }
We write a  A, 5  B, d
Is 0  C ? ___________
 A, or 7  B -- to indicate whether the object is an element or not an element
Is there a smallest member of C ? _____________
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Sets of real numbers:
N = set of natural numbers = set of counting numbers = set of positive integers
= { x | x  set of natural numbers } = { 1, 2, 3, 4, .... }
ex. Let A = { x | x represents the number of full time jobs that a student in this class has held }
Can this set be described by members of the set of natural numbers ?
ex. Let B = { y | y represents the number of days that a student will attend class during the semester, of those currently
in class (today) }
W = set of whole numbers = set of nonnegative integers = set of natural numbers and zero
= { x | x  set of whole number } = { 0, 1, 2, 3, ... }
ex. Let C represent the set that contains the number of times a student at ASU has enrolled in Math 1312
ex. Let D = represent the number of ASU graduates that are hired each year by a Fortune 500 company.
I = set of integers = set of whole numbers and their opposites
= { x | x  set of integers } = { ... -3, -2, -1, 0, 1, 2, 3, ... }
ex.
A game costs $1 to play and if you win, you will win $20.
Let C represent the amount of money that a player has “won” after playing n games. Can this amount be
represented by a whole number ? How about an integer ?
ex. In football a running back can either gain or lose yards. Let y represent the number of yards a runner has “gained”.
Can y be represented by an integer ?
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 ) , ...
ex. a recipe calls for a mixture that consists of a 30: 1 ratio of flour to sugar (cups). Can the a amount of sugar
be written as an integer ?
ex. your GPA, your car’s average mile per gallon, ...
Q / = set of irrational numbers → real numbers that can not be written in terms of a fraction
π, e,
ex.
2 , 0.1010010001... , ...
You are interested in finding a number so that when squared you will end up with 7.
found that satisfied that ?
Can a rational number be
ex. In order to measure the amount of fluid that a cylindrical container can hold the formula V = πr 2h is used.
Can you use a rational number to measure the volume ?
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R = set of real numbers : consists of the set of rational and irrational number – no other number
--- we can extend to an additional set that is called the set of complex numbers – our use of numbers will end with
the idea of real numbers
P = set of prime numbers – consist of natural numbers that are greater than 1 and can be evenly divided only by 1 and the
number itself
ex. A six-sided die is rolled. The faces are labeled with the first six prime numbers. List all possible values that
could occur.
If this die is rolled twice and the sum of the faces are added, what is the minimum and maximum sum possible ?
Examples of Sets: Identify each of the following as “good” sets or not
____________
1) Set of all female students in this class .
_____________ 2) the set of all companies with smart chairman
_____________ 3) set of all companies with executives who were given a salary exceeding 10million dollars / year
_____________ 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. 1.3 The set that contains no object is called the _____________ set of the _______ set.
Let E = set of all students in class with less than 1 member in the family
Let F = set of all students in class under 18 years of age that voted legally in the last election.
The number of times that the sum is greater than 12 on the roll of a pair of normal six sided dies
ex. Find all x’s so that
A = { x | x2 = 4 }
B={x|x<0 }
C = prime numbers
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Def. 1.4
The set that contains all objects under consideration is called the ______________________
-----Depending on the Experiment – this set will vary.
ex. Find all x’s so that
A = { x | x2 = 4 } if U = { x : x is a whole number }
A = ____________________
B = { x | x < 0 } and U = set of integers
B = ______________
C = set of prime numbers and U = set of even whole numbers → C = __________________
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. 1.5: Subsets
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.
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ex. A = set of all students enrolled in this class
B = set of all students enrolled at ASU
Which one is true
AB
BA
neither ?
_____________
ex. B = set of all companies that outsource jobs outside the company
C = set of all companies that made a profit last quarter
ex.
A = set of all students that drive to school
B = set of all students that arrive late to class
Venn Diagram of subsets: A  B.
Def.1.6: Equality
Let A and B be any two sets of some universal set U. We say that A and B are __________ provided A  B and B  A.
Ex. Let E = { x | x2 = 16 } and F = { x | x + 2 = 6 } → Are E and F equal ?
ex. Define A = set of al vowels and B = { a, e, i, o }
Is B  A ? Why ? __________________________________
Is A  B ? Why ? _________________________________________
Is A = B ? ___________________________________________
Conclude: If a set is not a subset of a second set,
there must be one element in the first set that is not in the second set.
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ex. It is obvious that {a, b }  { a, a, b } but what about { a, a, b }  { a, b } ?
what about { a, b }  { b, a } ?
Conclude that 1. 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
Counting Subsets:
We want to be able to count the elements of a set as well as subsets of a set ( possible outcomes)
Note:
Let A = { a, b, c }, is { } a subset of A ? ( Is   A ? )
Why or why not ?
A hamburger is to be made consisting of meat and buns – and a choice of
cheese, ketchup, onions, tomatoes, lettuce, pickles
One good question might be how many different hamburgers are possible ? ____________
Is a hamburger with only meat and buns possible ? __________
To what set does this hamburger correspond ? ________
Is a hamburger with everything on it possible ? __________
To what set does this hamburger correspond ? ________
ex.
The set A = { a } has one element and _____________ subsets (how many ? )
The set {1, 2 } has two elements and ____________ subsets ( how many ? )
The set {a, b, c } has three elements and ___________ subsets (how many ? )
The set { 1, 2, 3, 4 } has four elements and _____________ subsets ( how many ? )
In general , how many subsets does the set with n elements have ?
{ a1, a2, a3, ... , an } → has ____________ subsets ( how many ? )
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Proper Subsets:
If A  B and A ≠ B, then B has at least one element that is not in the set A.
ex.
A = { a, e, i, o, u } and B = {a, e, i, o, u, y } → A  B, but y  B and y  A → A ≠ B.
Whenever this happens we say that A is a proper subset of B.
Note:
A  B means that A is a subset of B and it could equal B ( similar to the idea of inequalities; x < 2, x ≤ 2 )
A  B means that A is a subset of B
We will use only the symbol  to represent both cases – when we want A to be a
proper subset, we will either say so or write “A  B, A ≠ B”.
Examples: of subsets
ex.
How many different pizzas can be made if the pizza has cheese plus a choice of
any of the following toppings; pepperoni, hamburger, sausage, or anchovies.
ex. The students in a business class write an essay about business ethics. The teacher grades the papers based on
grammatical mistakes. If there are no mistakes, the student gets a grade of 100. There are ten students in class.
How many different groups of students could get a grade of 100 ?
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 students that are at work right now and the set of all
students that are in bed sleeping.
How are these two sets related ?
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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
Could these be disjoint
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
ex.
A = The set of all employees that got a raise
B = { x : x is an odd whole number }
D = set of all non-positive integers
B = the set of all employees that got promoted
Disjoint sets ? Explain .
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.
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ex. Let U = { x : x is a whole number } with A = { x : x is positive } ,
B = { x : x is even whole number }
Find A/ = ____________________________
B / = ______________________________
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/ =
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Def. 1.9( Union:  and Intersection:  )
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
i.e. if x  A and x  B, then x  A ∩ B. or If x  A ∩ B, then x must be in A and x must be in B.
ex. A couple is planning to get married and they want to invite only the friends they have in common.
G: groom’s friends , B: bride’s friends → G = { a, b, c, d, e } and B = { a, d, e, f, g }
Who will get invited ?
C = set of people that get invited = _________________________________
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.
Thinking of the same couple from the previous example:
They will invite every friend they have – even if only one of them knows them.
Who will get invited ?
ex. A married couple decides to list all items that they own – is this a union or an intersection of two
sets?
Describe the two sets and the relationship between them.
Venn Diagrams of
A ∩ B
AB
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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
ex. of use of the word “or”
A = set of all customers that company A does business with
B = set of al customers that company B does business with
Let C represent the set of customers that either A or B does business with ( at least one of A or B does business with –
maybe both of them do )
Write C in set notation in terms of A and B. C = ________________
ex. of use of the word “and”
In the example above, company A and B will merge and they wish to eliminate duplications. Let C represent the
customers that A and B both do business with. Write C in set notation in terms of A and B.
C = ___________
ex. disjoint with symbols ∩ and  ( if A = ___ and B = ___, are A and B disjoint
when A  B = _____
A ∩ B = _____________
A  B = ____________
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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 U = set of all whole numbers let A = { x | x 2 = 4 } and B = set of all irrational numbers
example: let the set U represent the students that voted in the last local elections.
and B = set of male students under 18 years of age at the time of the last 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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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.
26
Sets and Counting(basic notation)
Ex. A company workforce consists of 45 men and 35 women. There are 16 smokers 12 of them male.
Write out sets A, B, C that describe the conditions (classifications) above.
Let A represent the set of ___________________________________________________
B
C
Once we define what sets A, B, and C are. We can shorten our notation to indicate how many objects are in the sets.
How many of the workers
a) are not smokers ? ________________
b) are either male or smokers ? _______________
c) are neither male nor smokers ? ____________________
ex. Let U = { x | x is a whole number } be the universal set with A = { x | x 2 < 4, x a whole number },
B = {x | x is a whole number less than 10 }, and C = set of all even prime numbers
n(A) represents the number of objects – the number of elements – in set A.
n(A) = _________,
n(A u B ) = ___________,
n(B) = _________,
n(C ) = ___________,
n(A n B ) = ___________
27
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 ? ______________
ex. A three letter word is to be made by selecting one letter from each of the following sets – in order.
{ a, b, c }
{ d, e }
{ f, g }
How many three letter words are possible ( they do not have to make sense ) ? _________________
ex. You are an assistant to a boss that always wants one of each of the following
a writing utensil: pencil, pen, marker
and one of three types of documents.
something to write on: printer paper, notebook
If he selects one of each , then how many different groups of three are possible ? _____________
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ex. 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
___________________________
To help us answer the questions above – we introduce the next and very important method(for counting purposes).
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 B = { 2, 3, 4 } . Find
Tree Diagram:
a
2
3
b
4
2
3
4
A x B = { _____________________________________________ },
B x A = { __________________________________________________ }
n ( A x B ) = _______________ n ( B 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 )
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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 ...) = _____________
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. An employee is classified according one of each of the following categories:
sex(m, f), department(1, 2, 3, 4 ), seniority( 0, 1, 2 ) .
How many distinct classifications are possible?
Solution:
Let A = { m, f ), B = { 1, 2, 3, 4 }, and C = { 0, 1, 2 } .
ex. A buffet dinner is served. Each guest is asked to select only one of each type –
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 ?
_________________
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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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Review of Sets:
Universal set, elements, subsets, equal sets, universal set, empty set, complement, # of
subsets, disjoint sets, union, intersection, cross product
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 A = { 1, 2, x } and B = { y, 3, 2 }, with B  A, then find the values of x and y. ___________
4. If x  E  F, then x must be where ? _______________________________ ( be complete)
5. If x  D ∩ E, then x must be where ? _______________________________ (be complete)
6. The ____________ of sets A and B consists of all objects that are in A or in B or in both A and B.
7. Which set is generally larger AB or A∩ B ? _________________
Is A ∩ B always a subset of A  B ? why or why not ? _____________________________________________
8.
If H = { x | x is the age of a student in class and it is known that every age from 18 to 25 is accounted for }
Find H ( list the members of H in set notation). H = __________________________
9. If A = { 1, 2, 3 } and B = { 2, 3, 4 }, then another name for the set { 2, 3 } would be ____________
10. If the universal set U = { 1, 2, 3, ..., 10 } and A = { x | x2 ≥ 4 }, then the complement of A = A / = ______________
11. Find the union of A = {a, e, i , o, u } and B = { b, c, d, f, g, h, .... } in set builder notation. ______________________
Find the intersection of A and B. ______________________________
12. Draw a Venn Diagram for each of the following sets. Make sure to include the universal set.
a) A  B/ =
b) A ∩ ( B ∩ C / )
13. 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.
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14. 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/ = 
15. How many subsets does the set { a, e, i, o, u } have ? ____________
How many of them are proper subsets ? __________
16. Two sets A and B are said to be equal provided 1) _____________________ and 2) ________________
17. Find the absolute value of each of the following
a) ___________________ =
18. 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 ?
19. Let A = { a, b, c, d } and B = { 1, 5 }. Find A x B.
20. If n( A ) = 5 and n(B ) = 4, then find n( A x B x A ) = __________
21. True or False.
___________ a) A x B = B x A .
__________ b) n( A x B ) = n (A ) n(B )
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22.
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 ?
23.
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 ?
24.
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 ?
34
Ch. 2
Basic Probability
I roll a die and the following values keep coming up ( 20 rolls)
2, 2, 1, 1, 1, 3, 4, 1, 1, 2, 2, 2, 4, 3, 4, 4, 1, 2, 1, 4
What kind of observations can you make ?
1) ( types of outcomes)
2) (sum of frequencies)
You continue rolling the die above 1000 times with the following results
( assume that the die behaves in a reasonable manner)
1: occurs 190 times
2: occurs 202 times
3: occurs 195 times
and 4: occurs 413 times
What do you conclude ?
1) (sum of frequencies)
2) ( fraction form )
3) (types of outcomes)
What would you expect to happen if you roll the die 50,000 times ?
What about n times ? ( as n gets very , very large ! )
Consider the same situations above with the toss of a single coin: (head)
As n gets “large” ( as n → ∞ ):
Consider the same situations above with the selection of a single card from a full deck of cards: (ace, two, ..., king )
As n → ∞ :
35
Instinctive Ideas of 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 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.
Half of each group is male. A person is selected at random. What is the likelihood that the person selected is
a female student ? ___________________
a male-management student ? ______________
36
8) What do you think the likelihood of guessing your birthday (day and month) correctly ? ____________
What if I have two tries ? _____________
What if I have six tries ? ______________
9) What is the probability that exactly two people in a class of 50 will have the same birthday ?
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.
37
Note:
Sets: universal set, elements, subset, disjoint ==>
Probability: sample space, sample points, 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 the set that contains all possible outcomes 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 ______________
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? ______
38
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
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.
39
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.
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
40
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 ? __________
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 ? _______
41
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 ? ________
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 ? ________
42
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 ) ?
43
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.
44
Odds in favor ( Of ) Odds against ( Og )
We define the odds in favor of an event E occurring as O f = 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/ ) = ______________________
45
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 / ) = ___________________
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Conditional Probability and Independent Events
Let A and B be any two events of some sample space S.
We write P( A | B ) to indicate the probability that event A will occur given that event B has known
to have already occurred.
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 ? ___________
47
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 )
Pf.
2) P ( A / ) = 1 – P( A ) , and
Pf.
*** 3) P( A  B ) = P( A )  P( A | B ),
We can simplify two of the formulas above
by having restrictions on the two given sets.
1) If A and B are mutually exclusive, then
P ( A  B ) = _______________
and 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 )
2) Independent
I will show you the result after some examples and explanation.
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.
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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.
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 ? _______________
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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 ? _______________
ex.
b) P ( A | B ) = _______
c) P( B | C ) = _______
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 ) = _______
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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.
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 ) = ________________________________________________________________________
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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. ________________
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ex. A class consists of 30 students; 20 male and 10 female. Six of the male and four female
students smoke. A student is selected at random. What is the probability that the student
a) smokes ? ________________
b) is a male student ? __________
Are the events smoking and being male mutually exclusive ? _________________
Are they independent ? _____________________________________________________
c) What is the probability that the students is male and smokes ? _________________
d) What is the probability that the student smokes if they are known to smoke ?
Suppose that the numbers were a little different
30 students; 24 male and 6 female → 4 male students smokes, 1 female student smokes
e ) are the events of smoking and being male independent ?
A card is drawn at random – card is replaced. What is the probability that an ace is drawn two
times in a row ( 2 draws ). ____________
A card is drawn at random – card is not replaced. What is the probability that an ace is drawn
two times in a row ( 2 draws). _________
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Bayes’s Formula.
ex. A diagnostic test is given to a group of students in an algebra at the beginning of the semester. 60 %
of the students pass the exam. If a student is known to have passed the diagnostic exam,
he has a 90 % chance of passing the class. If a student fails the diagnostic exam, he has a 50 %
chance of passing the class.
A student is selected at random. What is the probability that the student
a) will pass the diagnostic test and pass the class ? _________________
b) will pass the class ? _______________
c) will pass the class if they are known to have passed the diagnostic test ? ________
d) passed the diagnostic test if they are known to have passed the class ? _________
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example:
A test is given to identify potential heart problem cases within the next year. The test is known to be
accurate 80 % of the time. If the person is diagnosed with a heart problem, there is a 90 % he will
develop a heart problem within the year. If the person is diagnosed as having no problem, there is a
40 % chance that the person will develop heart problems within the year.
A person that that took this test is selected at random.
a) What is the probability that the person will develop a heart problem during the year ? ________
b) What is the probability that the test diagnosed the person correctly ? ________________
c) What is the probability that the test diagnosed somebody as having a heart problem if they are
known to have had a heart problem during the year.
____________
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ex. A group of 20 students are enrolled in class. 14 of them finish the semester with a grade of C or
better. A survey is done by selecting one of these students at random. If the student finished with a
C, the probability they found the course acceptable is 0.8. Otherwise the probability the student found
the course acceptable 0.40
Find each of the following probabilities.
A student is selected at random. What is the probability that the student
a) finished with a C or better ? _____________
b) found the course acceptable if they did not finish with a C or higher ? ___________
c) Finished and found the course acceptable ? ___________
d) Found the course acceptable ? ______________
e) f
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