statistics for business and economics chapter 3 probability
TRANSCRIPT
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Statistics for Business and Economics
Chapter 3
Probability
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Contents1. Events, Sample Spaces, and Probability2. Unions and Intersections3. Complementary Events4. The Additive Rule and Mutually Exclusive
Events
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Learning Objectives
1. Develop probability as a measure of uncertainty
2. Introduce basic rules for finding probabilities
3. Use probability as a measure of reliability for an inference
4. Provide an advanced rule for finding probabilities
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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?
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Many Repetitions!*
Number of Tosses
Total Heads Number of Tosses
0.00
0.25
0.50
0.75
1.00
0 25 50 75 100 125
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3.1
Events, Sample Spaces,and Probability
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Experiments & Sample Spaces
1. Experiment• Process of observation that leads to a single
outcome that cannot be predicted with certainty
2. Sample point• Most basic outcome of an
experiment
3. Sample space (S) • Collection of all sample points
Sample Space Depends on Experimenter!
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Visualizing Sample Space
1. Listing for the experiment of tossing a coin once and noting up face
S = {Head, Tail}
Sample point
2. A pictorial method for presenting the sample space Venn Diagram
HT
S
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Example
• Experiment: Tossing two coins and recording up faces:
• Is sample space as below?
S={HH, HT, TT}
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Tree Diagram
1st coin
H T
H T H T
2nd coin
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Sample Space Examples
• Toss a Coin, Note Face {Head, Tail}• Toss 2 Coins, Note Faces {HH, HT, TH, TT}• Select 1 Card, Note Kind {2♥, 2♠, ..., A♦} (52)• Select 1 Card, Note Color {Red, Black} • Play a Football Game {Win, Lose, Tie}• Inspect a Part, Note Quality {Defective, Good}• Observe Gender {Male, Female}
Experiment Sample Space
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Events
1. Specific collection of sample points
2. Simple Event
• Contains only one sample point
3. Compound Event
• Contains two or more sample points
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S
HH
TT
THHT
Sample Space S = {HH, HT, TH, TT}
Venn Diagram
Outcome;
Sample point
Experiment: Toss 2 Coins. Note Faces.
Compound Event: At least one Tail
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SHH
TTTH
HT
Sample Space S = {HH, HT, TH, TT}
Venn Diagram
Experiment: Toss 2 Coins. Note Faces.
Compound Event: Exactly one head
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SHH TT
TH
HT
Sample Space S = {HH, HT, TH, TT}
Venn Diagram
Experiment: Toss 2 Coins. Note Faces.
Compound Event: Tail at the 2nd toss
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SHH
TT
THHT
Sample Space S = {HH, HT, TH, TT}
Venn Diagram
Experiment: Toss 2 Coins. Note Faces.
Simple Event: Tail for both tosses
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Event Examples
• 1 Head & 1 Tail HT, TH
• Head on 1st Coin HH, HT
• At Least 1 Head HH, HT, TH
• Heads on Both HH
Experiment: Toss 2 Coins. Note Faces.
Sample Space: HH, HT, TH, TT
Event Outcomes in Event
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Thinking challenge
• A fair coin is tossed till to get the first head or four tails in a row. Which one is the sample space for this experiment?
a. S={T, TH, TTH, TTTH, TTTT}
b. S={T, HT, TTH, TTTH, TTTT}
c. S={H, TH, TTH, TTTH, TTTT}
d. S={H, HT, HHT, HHHT, HHHH}
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Probabilities
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What is Probability?
1. Numerical measure of the likelihood that event will occur
• P(Event)• P(A)• Prob(A)
2. Lies between 0 & 1
3. Sum of probabilities for all sample points in the sample space is 1
11
.5 .5
00
CertainCertain
ImpossibleImpossible
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Probability Rulesfor Sample Points
Let pi represent the probability of sample point i.
1. All sample point probabilities must lie between 0 and 1 (i.e., 0 ≤ pi ≤ 1).
2. The probabilities of all sample points within a sample space must sum to 1 (i.e., pi = 1).
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Equally Likely Probability
P(Event) = X / T• X = Number of outcomes in the
event
• 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.
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Steps for Calculating Probability
1. Define the experiment; describe the process used to make an observation and the type of observation that will be recorded
2. List the sample points
3. Assign probabilities to the sample points
4. Determine the collection of sample points contained in the event of interest
5. Sum the sample points probabilities to get the event probability
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Thinking Challenge
• Consider the experiment of tossing two balanced dice. Which of the following are true for this experiment.
I. The probability of having 4 or less for the sum of the dots on the up faces is 1/6.
II. The probability of having sum of the dots on the upfaces larger than 4 is 5/6
III.The probability of having 6 or less for the sum of the dots on the up faces is 7/12
a. I and II b. I and III c. II and III d. all
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Thinking Challenge (sol.)• Sample space for tossing two dice
S={(1,1),(1,2),…(1,6),(2,1),…,(2,6),…,(6,6)}
with 36 sample points
• Let event A=Having 4 or less for the sum of upfaces. So,
• A={(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)}
• Prob:{1/36,1/36,1/36,1/36,1/36,1/36}
• P(A)=6/36=1/6
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Thinking Challenge (sol.)• Let event B=Having the sum of upfaces
larger than 4. So,
• B={(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,2),(3,3),…(3,6),(4,1),…(4,6),(5,1),…(5,6),(6,1),…(6,6)}
With 30 sample points each with prob.1/36
• P(B)=30/36=5/6
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Thinking Challenge (sol.)• Let event C=Having the sum of upfaces 6 or
less. So,
• C={ (1,1), (1,2), (1,3), (1,4), (1,5), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2),(3,3), (4,1), (4,2), (5,1)}
each with prob.1/36
• P(C)=15/36=5/12
So the correct choice is a. I and II
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Combinations RuleA sample of n elements is to be drawn from a set of N elements. The, the number of different samples possible
is denoted byN
n
and is equal to
N
n
N!
n! N n !where the factorial symbol (!) means that
n!=n*(n-1)*…*3*2*1
5!54 321For example, 0! is defined to be 1.
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Thinking Challenge
• The price of a european tour includes four stopovers to be selected from among 10 cities. In how many different ways can one plan such a tour if the order of the stopovers does not matter?
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P(A)=0.3, P(B)=0.2
P(A)=0.25, P(B)=0.3
Ex.3.2 from text book (p.139)
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Ex.3.9 from text book
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a. S={Brown, yellow, red, blue, orange,green}
b. P={0.13, 0.14, 0.13, 0.24, 0.20, 0.16}
c. Let event A=selecting brown candy
P(A)=P(Brown)=0.13
d. Let event B=selecting red, green or yellow candy
P(B)= 0.13+0.16+0.14=0.43
e. Let event C= selecing a candy other than blue
P(C) = 0.13 +0.14+ 0.13+ 0.20+ 0.16=0.76
or P(C) = 1-0.24=0.76
Ex.3.9 from text book: solution
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Ex.3.25 from text book
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• Odds in favor of E= P(E) / [1-P (E)]
• Odds against E = [1-P (E)] / P(E)
a. Odds in favor of Oxford Shoes winning=(1/3)/(2/3)=1/2 meaning 1 to 2
b.1/1= P(E) / [1-P (E)] P(E)=1/2
c. Odds against Oxford Shoes winning = 3/2
3/2 = [1-P (E)] / P(E) P(E)=2/5
Ex.3.25 from text book: solution
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3.2
Unions and Intersections
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Compound Events
Compound events:
Composition of two or more other events.
Can be formed in two different ways.
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Unions & Intersections
1. Union• Outcomes in either events A or B or both• ‘OR’ statement• Denoted by symbol (i.e., A B)
2. Intersection• Outcomes in both events A and B• ‘AND’ statement• Denoted by symbol (i.e., A B)
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S
BlackAce
Event Union: Venn Diagram
Event Ace Black:
A, ..., A, 2, ..., K
Event Black:
2,
2,...,
A
Sample Space:
2,2,
2, ..., A
Event Ace:
A, A, A, A
Experiment: Draw 1 Card. Note Kind, Color & Suit.
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EventAce Black:
A,..., A, 2, ..., K
Event Union: Two–Way Table
Sample Space (S):
2, 2,
2, ..., A
Simple Event Ace:
A,
A,
A,
A
Simple Event Black:
2, ..., A
Experiment: Draw 1 Card. Note Kind, Color & Suit. Color
Type Red Black TotalAce Ace &
RedAce &Black
Ace
Non &Red
Non &Black
Non-Ace
Total Red Black S
Non-Ace
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S
BlackAce
Event Intersection: Venn Diagram
Event Ace Black:
A, A
Event Black:
2,...,A
Sample Space:
2, 2,
2, ..., A
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Event Ace:
A, A, A, A
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Sample Space (S):
2, 2,
2, ..., A
Event Intersection: Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Event
Ace Black:
A, A
Simple Event Ace:
A, A,
A, A
Simple Event Black: 2, ..., A
ColorType Red Black Total
Ace Ace &Red
Ace &Black
Ace
Non &Red
Non &Black
Non-Ace
Total Red Black S
Non-Ace
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Compound Event Probability
1. Numerical measure of likelihood that compound event will occur
2. Can often use two–way table• Two variables only
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EventEvent B1 B2 Total
A1 P(A 1 B1) P(A1 B2) P(A1)
A2 P(A 2 B1) P(A2 B2) P(A2)
P(B1) P(B2) 1
Event Probability Using Two–Way Table
Joint Probability Marginal (Simple) Probability
Total
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ColorType 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
Two–Way Table Example
Experiment: Draw 1 Card. Note Kind & Color.
P(Ace)
P(Ace Red)P(Red)
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Example from text book (p.145)
c. 0.16+0.03=0.19
0.10+0.16+0.03+0.10+0.08+0.22+0.14=0.83
P(A)=0.10+0.16+0.03=0.29
P(B)= 0.16+0.03+0.10+0.08+0.22+0.14=0.73
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Thinking Challenge
1. P(A) =
2. P(D) =
3. P(C B) =
4. P(A D) =
5. P(B D) =
EventEvent C D Total
A 4 2 6
B 1 3 4
Total 5 5 10
What’s the Probability?
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Solution*
The Probabilities Are:
1. P(A) = 6/10
2. P(D) = 5/10
3. P(C B) = 1/10
4. P(A D) = 9/10
5. P(B D) = 3/10
EventEvent C D Total
A 4 2 6
B 1 3 4
Total 5 5 10
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3.3
Complementary Events
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Complementary Events
Complement of Event A• The event that A does not occur• All events not in A• Denote complement of A by AC
S
AC
A
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Rule of Complements
The sum of the probabilities of complementary events equals 1:
P(A) + P(AC) = 1
S
AC
A
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S
Black
Complement of Event Example
Event Black:
2, 2, ..., A
Complement of Event Black,
BlackC: 2, 2, ..., A, A
Sample Space:
2, 2,
2, ..., A
Experiment: Draw 1 Card. Note Color.
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Back to M&M example
a. S={Brown, yellow, red, blue, orange,green}
b. P={0.13, 0.14, 0.13, 0.24, 0.20, 0.16}
c. Let event A=selecting brown candy
P(A)=P(Brown)=0.13
d. Let event B=selecting red, green or yellow candy
P(B)= 0.13+0.16+0.14=0.43
e. Let event C= selecing a candy other than blue
P(C) = 0.13 +0.14+ 0.13+ 0.20+ 0.16=0.76
or P(C) = 1-0.24=0.76 P(Cc) = 0.24
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3.4
The Additive Rule and Mutually Exclusive Events
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Mutually Exclusive Events
• Events do not occur simultaneously
• A does not contain any sample points
Mutually Exclusive Events
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S
Mutually Exclusive Events Example
Events and are Mutually Exclusive
Experiment: Draw 1 Card. Note Kind & Suit.
Outcomes in Event Heart:
2, 3, 4,
..., A
Sample Space:
2, 2,
2, ..., A
Event Spade:
2, 3, 4, ..., A
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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)
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Additive Rule Example
Experiment: Draw 1 Card. Note Kind & Color.
P(Ace Black) = P(Ace) + P(Black) – P(Ace Black)
ColorType Red Black Total
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52
52 52 52 52 4 26 2 28
= + – =
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Thinking Challenge
1. P(A D) =
2. P(B C) =
EventEvent C D Total
A 4 2 6
B 1 3 4
Total 5 5 10
Using the additive rule, what is the probability?
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10 10 10 10 6 5 2 9
Solution*
Using the additive rule, the probabilities are:
P(A D) = P(A) + P(D) – P(A D)1.
2. P(B C) = P(B) + P(C) – P(B C)
10 10 10 10
4 5 1 8
= + – =
= + – =
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a. AB
b. Ac
c. BC
d. Ac Bc
Exercise from text book (p.152)
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Example from text book (p.153)
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Let events
• F= being a fully compensated worker
• R= being a partially compensated worker
• N=being a noncompasated volunteer
• L= leaving because of retirement
a. P(F)=127/244
b. P(FL)= 7/244
c. P(Fc)=1-(127/244)=117/244
d. P(FL)=P(F)+P(L)-P(FL)=(127/244)+(28/244)-(7/244)=148/244
Example from text book (p.153): solution