Transcript pps

Statistics for Business and
Economics
Chapter 3
Probability
Learning Objectives
1. Define Experiment, Outcome, Sample Point,
Sample Space, Event, & Probability
2. Use a Venn Diagram, Two–Way Table, or
Tree Diagram to Find Probabilities
3. Describe & Use Probability Rules
Thinking Challenge
• What’s the probability
of getting a head on
the toss of a single fair
coin? Use a scale from
0 (no way) to 1 (sure
thing).
• So toss a coin twice.
Do it! Did you get one
head & one tail?
What’s it all mean?
Many Repetitions!*
Total Heads
Number of Tosses
1.00
0.75
0.50
0.25
0.00
0
25
50
75
Number of Tosses
100
125
Events, Sample Spaces,
and Probability
Experiments & Sample Spaces
1. Experiment
• Process of obtaining an observation, outcome or
simple event
2. Sample point
• Most basic outcome of an
experiment
Sample Space
Depends on
Experimenter!
3. Sample space (S)
• Collection of all possible outcomes
Sample Space Properties
1. Mutually Exclusive
•
2 outcomes can not
occur at the same
time
Experiment: Observe Gender
— Male & Female in
same person
2. Collectively Exhaustive
•
One outcome in
sample space must
occur.
— Male or Female
© 1984-1994 T/Maker Co.
Visualizing
Sample Space
1.
Listing
 S = {Head, Tail}
2.
Venn Diagram
H
T
S
Sample Space Examples
•
•
•
•
•
•
•
Experiment
Sample Space
Toss a Coin, Note Face
Toss 2 Coins, Note Faces
Select 1 Card, Note Kind
Select 1 Card, Note Color
Play a Football Game
Inspect a Part, Note Quality
Observe Gender
{Head, Tail}
{HH, HT, TH, TT}
{2♥, 2♠, ..., A♦} (52)
{Red, Black}
{Win, Lose, Tie}
{Defective, Good}
{Male, Female}
Events
1. Any collection of sample points
2. Simple Event
• Outcome with one characteristic
3. Compound Event
• Collection of outcomes or simple events
• Two or more characteristics
• Joint event is a special case
— Two events occurring simultaneously
Venn Diagram
Experiment: Toss 2 Coins. Note Faces.
Sample Space S = {HH, HT, TH, TT}
TH
Outcome
HH
Compound
Event: At
least one
Tail
HT
TT
S
Event Examples
Experiment: Toss 2 Coins. Note Faces.
Sample Space: HH, HT, TH, TT
•
•
•
•
Event
1 Head & 1 Tail
Head on 1st Coin
At Least 1 Head
Heads on Both
Outcomes in Event
HT, TH
HH, HT
HH, HT, TH
HH
Probabilities
What is Probability?
1. Numerical measure of the
likelihood that event will
cccur
• P(Event)
• P(A)
• Prob(A)
1
Certain
.5
2. Lies between 0 & 1
3. Sum of sample points is 1
0
Impossible
Probability
P(Event) = X / T
• X = Number of event outcomes
• T = Total number of sample points
in Sample Space
• Each of T sample points is equally
likely
— P(sample point) = 1/T
© 1984-1994 T/Maker Co.
Compound Events
Compound Events
Compound
Events
A or B
A and B
Not A
A given B
Unions & Intersections
Compound Events
Compound
Events
A or B
A and B
Not A
A given B
Unions & Intersections
1. Union
•
•
•
Outcomes in either events A or B or Both
‘OR’ statement
 symbol (i.e., A  B)
2. Intersection
•
•
•
Outcomes in both events A and B
‘AND’ statement
 symbol (i.e., A  B)
Event Union:
Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color &
Suit.
Sample
Space:
2R, 2R,
2B, ..., AB
Ace
Event Ace:
AR, AR, AB, AB
Black
S
Event
Black:
2B,
2B, ...,
AB
Event Ace  Black:
AR, ..., AB, 2B, ..., KB
Event Union:
Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color &
Suit.
Color
Sample Space
Type
(S):
Ace
2R, 2R,
2B, ..., AB
Non-Ace
Total
Ace & Ace & Ace
Red
Black
Non & Non & NonRed
Black Ace
Red
Black
S
Total
Event
Ace  Black:
AR,..., AB, 2B, ..., KB
Red
Black
Simple Event Black:
2B, ..., AB
Simple
Event
Ace:
AR,
AR,
AB,
AB
Event Intersection:
Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color &
Suit.
Sample
Space:
2R  , 2 R  ,
2B, ..., AB
Ace
Event Ace:
AR, AR, AB, AB
Black
S
Event Ace  Black:
AB, AB
Event
Black:
2B, ...,
AB
Event Intersection:
Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color &
Suit.
Color
Sample Space
Type
(S):
Ace
2R, 2R,
2B, ..., AB
Non-Ace
Event
Ace  Black:
AB, AB
Total
Total
Ace & Ace & Ace
Red
Black
Non & Non & NonRed
Black Ace
Red
Black
S
Red
Black
Simple
Event
Ace:
AR,
AR,
AB,
AB
Simple Event Black: 2B, ..., AB
Compound Event Probability
1. Numerical measure of likelihood that
compound event will occur
2. Can often use two–way table
• Two variables only
3. Formula methods
• Additive rule
• Conditional probability formula
• Multiplicative rule
Event Probability Using
Two–Way Table
Event
Event
B1
B2
Total
A1
P(A 1  B1) P(A1  B2) P(A1)
A2
P(A 2  B1) P(A2  B2) P(A2)
Total
Joint Probability
P(B 1)
P(B 2)
1
Marginal (Simple) Probability
Two–Way Table Example
Experiment: Draw 1 Card. Note Kind, Color &
Suit.
Color
Type
Red
Black
Total
Ace
2/52
2/52
4/52
Non-Ace
24/52
24/52
48/52
Total
26/52
26/52
52/52
P(Red)
P(Ace  Red)
P(Ace)
Thinking Challenge
What’s the Probability?
1. P(A) =
2. P(D) =
Event
C
D
4
2
3. P(C  B) =
Event
A
Total
6
4. P(A  D) =
B
1
3
4
5. P(B  D) =
Total
5
5
10
Solution*
The Probabilities Are:
1. P(A) = 6/10
2. P(D) = 5/10
Event
C
D
4
2
3. P(C  B) = 1/10
Event
A
Total
6
4. P(A  D) = 9/10
B
1
3
4
5. P(B  D) = 3/10 Total
5
5
10
Complementary Events
Compound Events
Compound
Events
A or B
A and B
Not A
A given B
Complementary Events
Complement of Event A
• The event that A does not occur
• All events not in A: AC
• P(A) + P(AC) = 1
AC
A
S
Complement of Event
Example
Experiment: Draw 1 Card. Note Kind, Color &
Suit.
Black
Sample
Space:
2R, 2R,
2B, ..., AB
Event Black:
2B, 2B, ..., AB
S
Complement of Event Black,
BlackC: 2R, 2R, ..., AR, AR
Additive Rule & Mutually
Exclusive Events
Compound Events
Compound
Events
A or B
A and B
Not A
A given B
Mutually Exclusive Events
Mutually Exclusive Events
• Events do not occur
simultaneously
• A  B does not contain
any sample points

Mutually Exclusive
Events Example
Experiment: Draw 1 Card. Note Kind & Suit.
Sample
Space:
2, 2,
2, ..., A


Event Spade:
2, 3, 4, ..., A
S
Outcomes
in Event
Heart:
2, 3, 4
, ..., A
Events  and are Mutually Exclusive
Additive Rule
1. Used to get compound probabilities for
union of events
2. P(A OR B) = P(A  B)
= P(A) + P(B) – P(A  B)
3. For mutually exclusive events:
P(A OR B) = P(A  B) = P(A) + P(B)
Additive Rule Example
Experiment: Draw 1 Card. Note Kind, Color &
Suit.
Color
Type
Ace
Red
Black
2
2
Total
4
Non-Ace
24
24
48
Total
26
26
52
P(Ace  Black) = P(Ace) + P(Black) – P(Ace  Black)
4
26
2
28
=
+
–
=
52
52 52 52
Thinking Challenge
Using the additive rule, what is the probability?
1. P(A  D) =
2. P(B  C) =
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Solution*
Using the additive rule, the probabilities are:
1. P(A  D) = P(A) + P(D) – P(A  D)
6
5
2
9
=
+
–
=
10 10 10 10
2. P(B  C) = P(B) + P(C) – P(B  C)
4
5
1
8
=
+
–
=
10 10 10 10
Conditional Probability
Compound Events
Compound
Events
A or B
A and B
Not A
A given B
Conditional Probability
1. Event probability given that another event
occurred
2. Revise original sample space to account for
new information
• Eliminates certain outcomes
3. P(A | B) = P(A and B) = P(A  B)
P(B)
P(B)
Conditional Probability Using
Venn Diagram
Ace
Black
S
Event (Ace  Black)
Black ‘Happens’:
Eliminates All
Other Outcomes
Black
(S)
Conditional Probability Using
Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color &
Suit.
Color
Type
Red
Black
Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Revised
Sample
Space
P(Ace  Black) 2 / 52
2
P(Ace | Black) =


P(Black)
26 / 52 26
Statistical Independence
1. Event occurrence does not
affect probability of another
event
• Toss 1 coin twice
2. Causality not implied
3. Tests for independence
• P(A | B) = P(A)
• P(A  B) = P(A)*P(B)
Thinking Challenge
Using the table then the formula, what’s the
probability?
1. P(A|D) =
2. P(C|B) =
3. Are C & B
Independent?
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Solution*
Using the formula, the probabilities Are:
P(A  D) 2 / 10 2
P(A | D) =
=
=
P(D)
5 / 10 5
P(C  B) 1 / 10 1
P(C | B) =
=
=
P(B)
4 / 10 4
5
1
≠ = P(C | B)
P(C) =
10 4
Dependent
Multiplicative Rule
Compound Events
Compound
Events
A or B
A and B
Not A
A given B
Multiplicative Rule
1. Used to get compound probabilities for
intersection of events
• Called joint events
2. P(A and B) = P(A  B)
= P(A)*P(B|A)
= P(B)*P(A|B)
3. For Independent Events:
P(A and B) = P(A  B) = P(A)*P(B)
Multiplicative Rule Example
Experiment: Draw 1 Card. Note Kind, Color &
Suit.
Color
Type
Ace
Red
Black
2
2
Total
4
Non-Ace
24
24
48
Total
26
26
52
P(Ace  Black) = P(Ace)∙P(Black | Ace)
 4  2  2
    
 52  4  52
Thinking Challenge
Using the multiplicative rule, what’s the
probability?
1. P(C  B) =
Event
C
D
4
2
2. P(B  D) =
Event
A
Total
6
3. P(A  B) =
B
1
3
4
Total
5
5
10
Solution*
Using the multiplicative rule, the probabilities
are:
P(C  B) = P(C)  P(B| C) = 5/10 * 1/5 = 1/10
P(B  D) = P(B)  P(D| B) = 4/10 * 3/4 = 3/10
P(A  B) = P(A)  P(B| A) = 0
Tree Diagram
Experiment: Select 2 pens from 20 pens: 14
blue & 6 red. Don’t replace.
Dependent!
P(R) = 6/20
P(R|R) = 5/19
R
P(R  R)=(6/20)(5/19) =3/38
P(B|R) = 14/19
P(R|B) = 6/19
B
R
P(R  B)=(6/20)(14/19) =21/95
R
P(B  R)=(14/20)(6/19) =21/95
B
P(B) = 14/20
P(B|B) = 13/19 B P(B  B)=(14/20)(13/19) =91/190
Bayes’s Rule
Bayes’s Rule
• Allows computation of an unknown
conditional probability, P(B|A), by converting
it to a known conditional probability, P(A|B)
• For k mutually exclusive events,
P( Bi ) P( A | Bi )
P( Bi | A) 
P( B1 ) P( A | B1 )  P( B2 ) P( A | B2 )  ...  P( Bk ) P( A | Bk )
Bayes’s Rule Example
A company manufactures mp3 players at two factories.
Factory I produces 60% of the mp3 players and
Factory II produces 40%. Two percent of the mp3
players produced at Factory I are defective, while 1%
of Factory II’s are defective. An mp3 player is selected
at random and found to be defective. What is the
probability it came from Factory I?
Bayes’s Rule Example
P(I) = .6
P(II) = .4
Factory
I
P(D|I) = .02
Defective
P(G|I) = .98
Good
P(D|II) = .01
Defective
P(G|II) = .99
Good
Factory
II
P( I ) P( D | I )
.6*.02
P ( I | D) 

 .75
P( I ) P( D | I )  P( II ) P( D | II ) .6*.02  .4*.01
Conclusion
1. Defined Experiment, Outcome, Sample
Point, Sample Space, Event, & Probability
2. Used a Venn Diagram, Two–Way Table, or
Tree Diagram to Find Probabilities
3. Described & Use Probability Rules