Transcript Ch 2A Sample Spaces and Events

Probability
Chapter 2
A random walk down a probabilistic
path leading to some stochastic
thoughts on chance events and
uncertain outcomes.
The world is an
uncertain place
Chapter 2A
Sections 2-1 thru 2-2
Let’s get
started!
Random Experiment

An experiment or process that can result in
different outcomes each time it is repeated
or observed.
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
tossing a pair of dice
observing time between failures of a
machine
counting the number of defects found per
1000 units produced
measuring the diameter of a wire produced
to specifications
inspecting the quality of a finished product
Basic Theme
Predicting future outcomes of random
processes can be very non-intuitive
What random processes
 Therefore a method (i.e. a
are you familiar with?
mathematical system) is needed to
quantify their likelihoods.

Some Random Processes
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The weather – is it going to rain?
Life – a birth – when? will it be a boy or a girl?
Death – when am I going to die?
Illness – will I catch the flu this season?
Exams– will I pass the prob/stat exam?
Sports – will UD win its next basketball game?
Lottery and gambling – will I win the Ohio lottery if I play
tomorrow?
Drug Testing – will I be selected? Will I test positive?
Investments – will I lose money this year on my stock
holdings?
Reliability – when is this car going to break down again?
Sample Space
The set of all possible outcomes of a random
experiment.
 Denoted by S.

Tossing a die: S = {1,2,3,4,5,6}
 Observing time to next failure: S = {x| x 0}
 Counting defects among 1000 parts
produced: s = {x|x = 0, 1, … , 1000}
 Inspecting quality of a finished product: S =
{good, mediocre, poor}

More about those sample spaces

A sample space is discrete if it consists of a finite (or
countably infinite) set of outcomes
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Manufactured part – Acceptable or Defective – finite
number of outcomes – discrete
Time recording device – records hours with two
decimal place accuracy (countably infinite).
A sample space is continuous if it contains an
interval of real numbers – whether the interval is
finite or not.
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The time taken for a light bulb to burn out.
The number of gallons of gasoline consumed to travel
a fixed distance
The package weight of a product coming off the final
production line
Events

a launch event
into the sample
space
A subset of a sample space


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Tossing a die: S = {1,2,3,4,5,6}; define
event E, getting an even number: E =
{2,4,6}
Observing time to next failure: S = {x| x 0};
define event E as the next failure occurs
within 24 hours: E = {x|x  24}
Counting defects among 1000 parts
produced: S = {x|x = 0, 1, … , 1000}; define
event E as no more then .5% defects: E = {x|
x = 0, 1, 2, 4, 5}
2-1 Sample Spaces and Events
Basic Set Operations
Need Help on Sets?
If you need a refresher on set theory, the
motivated student is encouraged to
download and read the set theory
eReserve material on the course
Website.
http://academic.udayton.edu/charlesebeling/ENM500/Admin/ereserve.htm
More on Events
Given a sample space S, an event E
is a subset of S.
 The outcomes in E are called the
favorable outcomes. We say that E
occurs in a particular experiment if
the outcome of that experiment is one
of the elements of E, that is, if the
outcome of the experiment is
favorable.

Basic set operations
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Assume an inspection process with the following
outcomes: Pass (P), Rework (R), Scrap (C)
Let E1 = {P, R} and E2 = {R, C}
Union ( ) – set of all outcomes contained in either
of two events E1 E2 = {P, R, C} = S
Intersection ( ) – set of all outcomes contained in
both events E1 E2 = {R}
Complement (E’) – set of all outcomes in the
sample space not in the event E


E1’ = {C}
S’ = { } = 
The Laws of the Algebra of Sets
These are
very good
laws.
De Morgan's Laws:
(A  B)' = A'  B‘
(A  B)' = A'  B‘
Idempotent Laws:
(A  A) = A
(A  A) = A
Associative Laws:
(AB)C = A(BC)
(AB)C = A(BC)
Commutative Laws:
(A  B) = (B  A)
(AB)= (BA)
Distributive Laws:
A(BC) = (AB)C)
A(BC) = (AB)C)
Identity Laws
(A  ) = A
(A  S) = S
(A  ) = 
(A  S) = A
Complement Laws
(A  A’) = S
(A’)’ = A
(A  A’) =  S’ = , ’ = S
Other set properties

Mutually exclusive – two events having no
common outcomes are said to be mutually
exclusive.

E1 E2 =  says that E1 and E2 are mutually
exclusive
Tree Diagram as Sample Space
Representation
A
A’
(A’,B’,C’)
C’
B’
C’
B
B’
B
C
Tips of the trees represent all
of the possible combinations of
outcomes.
C
C’
C
C’
C
2-1 Sample Spaces and Events
Figure 2-5 Tree diagram for three messages.
Venn Diagrams
Shading is often used
to highlight the event
of interest.
Union

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
A union of two or more events is another event that contains
all outcomes contained in the previous events.
Union is designated by the symbol . If A and B are events
then A  B represents the union of A and B
The union of A and B is the set of all outcomes that are
either in A or B (or both), therefore
A  B = {x | x  A or x  B}.
“such that”
“OR” logic
Intersection
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The intersection of two (or more) events are those outcomes
that they have in common.
Intersection is designated by the symbol . So if A and B are
events then the intersection is denoted by A  B.
The intersection of A and B is the set of all outcomes that are
common to A and B, therefore
A  B = {x | x  A and x  B}
“And” logic
Mutual Exclusive Events
Two event are mutually exclusive (also called disjoint) if they do
not have any elements in common
AB=
Complement
Given the event A, then the event A' is the
complement of A consisting of all elements
not in A; i.e. A’ = {x| x  A}
A
A’
S
Let S = the sample space (the set of all events under discussion)
Then A  A’ = S and A  A’ =  (they are mutually exclusive)
Sometimes the complement of an event A is written as Ac
An Example
An engineer is selected at random within a large
manufacturing company. S = {all engineers of a
large manufacturing company}. Define
P = event, engineer selected was promoted
M = event, engineer selected has a Master’s degree
Find the event:
1. Engineer was promoted and has a Master’s degree.
2. Engineer was not promoted and did not have a
Master’s degree.
3. Engineer was either promoted or has a Master’s
degree.
4. Engineer did not get promoted but does have a
Master’s degree
5. Engineer did not get promoted or has a Master’s
degree.
PM
Pc  Mc
PM
Pc  M
Pc  M
Yet Another Example
S = {all students taking prob/stat, org behavior, and eng. economy}
A = event, a student passes the prob/stat course
B = event, a student passes the org behavior course
C = event, a student passes the engineering economy course
Describe the event:
1. A student passes all 3 courses
ABC
ABC
2. A student passes at least one of the courses
3. A student passes only prob/stat
A  Bc  Cc
4. A student fails org behavior
Bc
5. A student fails at least two courses
(Ac  Bc  C)  (Ac  B  Cc)  (A  Bc  Cc)  (Ac  Bc  Cc)
To recap – Random Processes
A process having an uncertain outcome called a
random event that occurs by chance and cannot be
predicted with certainty.
The set of all possible random events is called the
sample space.
We are counting on you!
Counting
Techniques
Combinatorics is a branch of mathematics that
studies collections (usually finite) of objects that
satisfy specified criteria. In particular, it is concerned
with "counting" the objects in those collections
(enumerative combinatorics).
2-1 Sample Spaces and Events
2-1.4 Counting Techniques
Multiplication Rule
Permutations
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A PERMUTATIONS of n objects, is all of
their possible arrangements or orderings.
For example, the permutation of the letters
a, b, and c is:


abc acb bac
bca cab cba
Of primary interest is knowing the number
of permutations that are available from a
collection of objects.
Permutations

To find the number of permutations of
four objects:
there are 4 ways to choose the first,
 3 ways remain to choose the second,
 2 ways to choose the third,
 and 1 way to choose the last.
 Therefore the number of permutations
of 4 different things is 4· 3· 2· 1 = 24

Permutations of less than all
A permutation of n things taken r at a time
is an ordered arrangement of r objects
selected from the n objects.
 Given the letters a, b, c then the
permutations of the 3 letters taken 2 at a
time are:


ab
ba ac ca bc cb
How many are there?
The number of permutations of n
objects taken r at a time is denoted by
P(n,r) or nPr or Pn,r.
n!
P(n, r )  n(n  1)(n  2)...(n  r  1) 
(n  r )!
2-1 Sample Spaces and Events
2-1.4 Counting Techniques
Permutations : Example 2-10
2-1 Sample Spaces and Events
2-1.4 Counting Techniques
Permutations of Similar Objects
Permutations with Repetitions
Example: How many distinct signals, each consisting of
6 flags hung in a line, can be formed from 4 identical red
flags, and 2 blue flags?
a
b
c
a
6!
 15
4!2!
d
b
2-1 Sample Spaces and Events
2-1.4 Counting Techniques
Permutations of Similar Objects: Example 2-12
Combinations
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A combination is an unordered selection of
objects.
A combination of n objects taken r at a time
is any selection of r of the objects where
order does not count.
The number of combinations of n objects
taken r at a time is denoted by
C (n, r ) n Cr  Cn ,r
n
n!
C  
 r  r !(n  r )!
n
r
The Number of
Combinations

Since an ordered arrangement of r objects
from among n objects is a permutation
where
n!
P(n, r ) 

(n  r )!
then to ignore the order of the r objects,
divide by r!, the number of ways in which
each set of r objects may be ordered.
P(n, r )
n!
C (n, r ) 

r!
r !(n  r )!
Proof by Example
Given 4 objects (a,b,c,d) taken 3 at a time
Combinations
abc
abd
acd
bcd
Permutations
abc, acb, bac, bca, cab, cba
abd, adb, bad, bda, dab, dba
acd, adc, cad, cda, dac, dca
bcd, bde, cbd, cdb, dbc, dcb
P(4,3)
4!
C (4,3) 

4
3!
3!(1)!
Some Combination Examples
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How many committees of 3 can be formed from 8
people?
 C(8,3) = 8! / 3! = (8)(7)(6) / [(1)(2)(3)] = 56
In a deck of 52 playing cards, how many different
poker hands of 5 cards are there?
 C(52,5) = 2,598,960
In a deck of 52 playing cards, how many ways
can a spade flush (5 spades) be dealt?
 C(13,5) = 1287
2-1 Sample Spaces and Events
2-1.4 Counting Techniques
Combinations: Example 2-13
2-2 Interpretations of
Probability
Measures
uncertainty
Foundation of inferential statistics
Basis for decision models under
uncertainty
What is a probability – P(E)?
Let P(E) =Probability of the event E occurring,
then
 0  P(E)  1
 If P(E) = 0, then event will not occur
(impossible event)
 If P(E) = 1, then the event will occur, i.e.
a certain event
So the closer P(E) is to 1,
then the more likely it is
that the event E will
occur?
Elementary or basic events
Three interpretations of probability:
1.
Empirical or relative frequency

2.
A priori or equally-likely

3.
Derived from historical data
Derived using counting methods
Subjective

Derived from personal judgment or
an individual’s degree of belief in an
outcome
Relative Frequency (empirical)
Let n(E) = number of times event E occurs in n
trials
n( E )
P ( E )  Lim
n 
n
Example - Relative Frequency
(empirical)

A coin is tossed 2,000 times and heads
appear 1,243 times.


The company’s Web site has been down 5
days out of the last month (30 days).


P(H) = 1243/2000 = .6215
P(site down) = 5/30 = .16667
2 out of every 20 units coming off the
production line must be sent back for
rework.

P(rework) = 2/20 = .1
A priori (equally-likely
outcomes)
A priori (knowable independently of experience):
n(A) = the number of ways in which event A can
occur
n(S) = total number of outcomes from the
random process
n( A)
P( A) 
n( S )
Example 1 - A priori (equally-likely
outcomes)

A pair of fair dice are tossed.
S = {(1, 1),(1, 2),(1, 3),(1, 4),(1, 5),(1, 6), (2, 1),(2, 2),(2,
3),(2, 4),(2, 5),(2, 6),(3, 1),(3, 2),(3, 3),(3, 4),(3, 5),(3,
6),(4, 1),(4, 2),(4, 3),(4, 4),(4, 5),(4, 6),(5, 1),(5, 2),(5,
3),(5, 4),(5, 5),(5, 6),(6, 1),(6, 2),(6, 3),(6, 4),(6, 5),(6,
6)}
P(a seven) = 6/36 = .1667
Example 2 - A priori (equally-likely
outcomes)

5 parts are selected from a bin containing 40 parts, 10 of
which are known to be defective. What is the probability of
no defects among the 5 parts selected?


How many ways can the 5 parts be selected? That is all many
outcomes in the sample space?
How may of these contain no defects? That is, all many favorable
outcomes?
Prob =
number favorable outcomes
total number of outcomes
 30  10 
  
5   0  142,506



 .21657
40
658, 008
 
 
5 
Subjective Probabilities

Subjective – personal degree of belief in the likelihood
of an outcome
You must buy now. This stock
has a 90 percent chance of
doubling in value within the
year.
Shady Investments Inc.
More Subjective Probabilities

Subjective – personal degree of belief in the likelihood
of an outcome
Believe me. This car is 99
percent reliable!
Probability of a failure = .01
Compound events

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Formed from unions, intersections, and
complements of basic events
Probabilities derived from


Axioms
Definitions
• Sample Space
• Events
• Conditional Probability

Theorems
• Addition law
• Multiplication law
• Bayes theorem

Collectively constitute the mathematics of
probability
2-2 Interpretations of Probability
2-2.2 Axioms of Probability
Mutually Exclusive Events
S = {(1, 1),(1, 2),(1, 3),(1, 4),(1, 5),(1, 6), (2, 1),(2, 2),(2, 3),(2, 4),(2, 5),(2, 6)
(3, 1), (3, 2),(3, 3),(3, 4),(3, 5),(3, 6),(4, 1),(4, 2),(4, 3),(4, 4),(4, 5),(4, 6),(5, 1)
(5, 2), (5, 3),(5, 4),(5, 5),(5, 6),(6, 1),(6, 2),(6, 3),(6, 4),(6, 5),(6, 6)}

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A pair of dice are tossed.
Find Pr{a seven is tossed or doubles are
tossed}
Let A = event, a seven is tossed
Let B = event, doubles are tossed
P(A) = 6/36 and P(B) = 6/36
P(A  B) = P(A) + P(B) = 12/36 = .3333
Next Time
Can I start on the
homework right a
way? And can I do
extra problems as
well?