1. Busn 210 Chapter 5 A Survey of Probability Concepts
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2. Goals Probability Concepts Define probability Understand the terms:
Busn 210 Chapter 5
Goals
Probability Concepts
Define probability
Understand the terms:
Experiment
Outcome
Event
Describe these approaches to probability:
Classical
Empirical
Subjective
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3. Goals Probability Concepts
Busn 210 Chapter 5
Goals
Probability Concepts
Calculate probabilities applying these rules:
Rules of addition
Rules of multiplication
Define the terms:
Conditional probability
Joint probability
Contingency Tables
Use a tree diagram to organize and compute probabilities
Principles of Counting
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4. Types Of Statistics Inferential Statistics
Busn 210 Chapter 5
Types Of Statistics
Inferential Statistics
A decision, estimate, prediction, or generalization about a population, based on a sample
(Second part of definition of statistics)
Also known as:
Statistical inference
Inductive statistics
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5. Statistical Inference Or Inferential Statistics
Busn 210 Chapter 5
Statistical Inference Or Inferential Statistics
Computing the chance that something will occur in the future!
This means that we will have to make decisions with incomplete information
Seldom does a decision maker have complete information from which to make a decision:
Marketers taking samples about a product name
Tests for wire tensile strength
Which player should the Mariners draft?
Should the soap opera Days of Our Lives be discontinued immediately?
Should I marry Jean?
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6. Busn 210 Chapter 5
Future Uncertainty
“Because there is uncertainty in decision making, it is important that all the known risks involved be evaluated scientifically”
Probability Theory will help:
Decision makers with limited information analyze the risks and minimize the inherent gamble
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7. Define Probability “Chance,” Likelihood,” Probability”
Busn 210 Chapter 5
Define Probability
“Chance,” Likelihood,” Probability”
A number between zero & one, inclusive, describing the relative possibility (chance or likelihood) an event will occur in the future
Decimal or fraction: .25 = ¼, etc.
0 ≤ P (x) ≤ 1
x means “the event”
P means probability
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8. Define Probability
A value of zero means it cannot happen
A value near zero means the event is not likely to happen
A value of one means it is certain to happen
A value near one means it is likely
Probability
Is the probability that a World Series will happen in 2006 close to one or to zero?
Is the probability that a company will name a new breakfast cereal “Crud That Hurts Your Tummy” close to one or to zero?

9. Understand The Terms: Experiment Outcome Event
Busn 210 Chapter 5
Understand The Terms:
Experiment
Doing something and observing the one result
Outcome
A particular result of the experiment
Event
A collection of one or more outcomes of an experiment
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10. Busn 210 Chapter 5
Define: Experiment
A process that leads to the occurrence of one and only one of several possible observations
Example: roll die & there are 6 possible outcomes
Ask 250 Highline students whether they drink coffee
An experiment has two or more possible results (outcomes), and it is uncertain which will occur
An experiment is the observation of some activity or the act of taking some measurement
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11. Define: Outcome An outcome is the particular result of an experiment
Busn 210 Chapter 5
Define: Outcome
An outcome is the particular result of an experiment
Examples:
When you toss a coin, the possible outcomes are:
Heads
Tails
When you survey 1000 people and ask whether they will vote for candidate 1 or candidate 2, some of the possible outcomes are:
455 would vote for candidate 1
592 would vote for candidate 1
780 would vote for candidate 1…
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12. Busn 210 Chapter 5
Define: Event
When one or more of the experiment’s outcomes are observed, we call this an event!
An event is the collection of one or more outcomes of an experiment
Example:
Roll die:
An even number can be an event
Boomerang tournament:
More than ½ the participants earned more than 60 points in the Trick Catch event
Political poll:
Less than 50% of those polled said they would vote for candidate A
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13. Experiment, Outcome, Event
Busn 210 Chapter 5
Experiment, Outcome, Event
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14. Definitions Sample Space
Busn 210 Chapter 5
Definitions
Sample Space
A representation (list) of all possible outcomes in an experiment
It can be hard to list all the outcomes
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15. Venn Diagram Event A P(A) =.2 Sample Space
Busn 210 Chapter 5
Venn Diagram
Sample Space
Event A
P(A) =.2
 A
Sample Space is all outcomes = P(x) = 1
Compliment P( A) = 1 – P(A) = 1 – .2 = .8
P(A) + P(~A) = 1 or P(A) = 1 - P(~A)
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16. Define: Mutually Exclusive
Busn 210 Chapter 5
Define: Mutually Exclusive
Ch 5 Mutually Exclusive:
The occurrence of one event means that none of the others can occur at the same time
Two events, A & B, are mutually exclusive if both events, A & B, cannot occur at the same time
In Venn Diagrams, there is no intersection:
Examples:
Die tossing experiment: the event “an even number” and the event “an odd number are mutually exclusive
If you get an odd, it cannot also be even
You can not have a product come off the assembly line that is both defective and satisfactory
Even
Odd
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17. Definitions Collectively Exhaustive
Busn 210 Chapter 5
Definitions
Collectively Exhaustive
At least one of the events must occur when an experiment is conducted
If an experiment has a set of events that include every possible outcome, such as the events “an even number” and “an odd number,” then the set of events is collectively exhaustive
Mutually Exclusive & Collectively Exhaustive
If a set of events is mutually exclusive & collectively exhaustive, then the sum of the probabilities are equal to 1
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18. Definitions Independent Dependent
Busn 210 Chapter 5
Definitions
Independent
Events are independent if the occurrence of one event does not affect the occurrence of another (sample space is not changed)
The roll of a six, does not affect the next roll
P(B│A) = P(B)
Dependent
Events are dependent if the occurrence of one event affects the occurrence of another event (sample space is changed)
The chances of pulling a heart from a deck of cards? 13/52. But if you don’t put the card back (without replacement), what is the probability that you pull a heart next time? It depends:
13/51 or 12/51
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19. Definitions Conditional Probability The sample space will change
Busn 210 Chapter 5
Definitions
Conditional Probability
The probability of a particular event occurring, given that another event has occurred
The sample space will change
The probability of the event B given that the event A has occurred is written P(B|A)
In the heart example, 13/51 or 12/51 are conditional probabilities
Line means “given that.”
“Probability that B will occur given that A has already occurred”
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20. Definitions Joint Probability
Busn 210 Chapter 5
Definitions
Joint Probability
A joint probability measures the likelihood that two or more events will happen concurrently
An example would be the event that a student has both a DVD Player and TV in his or her dorm room
Sample Space
Has a DVD player
Has a TV
Root probabilities times conditional probabilities equal joint probabilities (Tree Diagrams)
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21. Busn 210 Chapter 5
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22. Classical Approach To Probability
Busn 210 Chapter 5
Classical Approach To Probability
The Classical definition applies when there are n equally likely outcomes
Each outcome must have the same chance of occurring (fairness)
Events must be mutually exclusive & collectively exhaustive
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23. Classical Approach To Probability
Busn 210 Chapter 5
Classical Approach To Probability
A fair die is rolled once.
The experiment is rolling the die.
The possible outcomes are the numbers 1, 2, 3, 4, 5, and 6.
An event is the occurrence of an even number. That is, we collect the outcomes 2, 4, and 6.
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24. Classical Approach To Probability
Busn 210 Chapter 5
Classical Approach To Probability
We do not need to conduct experiments to determine the probability under the classical approach? No!
Cards, dice, taxes
Example:
If three million returns are sent to your district office and 3000 will be audited the probability that you will be audited is:
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25. Empirical Approach To Probability
Busn 210 Chapter 5
Empirical Approach To Probability
The empirical definition applies when the number of times the event happens (in past) is divided by the number of observations
The probability of an event happening is the fraction of the time similar events happened in the past.
“Relative Frequency”
Law of Large Numbers: Over a large number of trials the empirical probability of an event will approach its true probability. This law allows us to use relative frequencies to make predictions.
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26. Empirical Approach To Probability
Busn 210 Chapter 5
Empirical Approach To Probability
Throughout her teaching career Professor Jones has awarded 186 A’s out of 1,200 students. What is the probability that a student in her section this semester will receive an A?
To find the probability a selected student will earn an A:
Based on past experience, we can estimate that the probability that a student will receive an A grade in a future class is .155
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27. Subjective Approach To Probability
Busn 210 Chapter 5
Subjective Approach To Probability
Subjective probability
There is little or no past experience on which to base probability
An individual assigns (estimates) a probability based on whatever information is available
Examples:
Estimate the probability that the Mariners will the World Series next year
Estimate the probability that AOL will merge with GOOGLE
Estimate the probability that a particular corporation will default on a loan
Estimate the probability mortgage rates will top 8 percent
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28. Probability P(x) is never known with certainty
Busn 210 Chapter 5
Probability
P(x) is never known with certainty
P(x) is an estimate of an event that will occur in the future
There is great latitude in the degree of uncertainty surrounding this estimate
The degree of uncertainty is primarily based on the knowledge possessed by the individual concerning the underlying process
We know a great deal about rolling die
The underlying process is straight forward
We may not know much about whether a merger between companies will occur
Only some parts of the underlying process are known
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29. Busn 210 Chapter 5
Probability
The same laws of probability will be used regardless of the level of uncertainty surrounding the underlying process
Individuals will assign probabilities to events of interest
The difference amongst them will be in their confidence in the precision of the estimate
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30. Busn 210 Chapter 5
Sample Space
Event A
Has a DVD player
Event B
Has a TV player
In this circumstance, Events A & B are not mutually exclusive!
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31. Calculate Probabilities Applying These Rules:
Busn 210 Chapter 5
Calculate Probabilities Applying These Rules:
Rules of addition
Rules of multiplication
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32. Rules Of Addition Why? Don’t want to count twice! P(X)  1
Busn 210 Chapter 5
Rules Of Addition
Event A
P(A)
Event B
P(B)
Why? Don’t want to count twice!
P(X)  1
HW #22, page 152 “at least one” = “either or”
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33. Rules Of Addition Mutually Exclusive! Event B Event A P(B) P(A)
Busn 210 Chapter 5
Rules Of Addition
Mutually Exclusive!
Event A
P(A)
Event B
P(B)
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34. Rules Of Addition Example 1
Busn 210 Chapter 5
Rules Of Addition Example 1
New England Commuter Airways recently supplied the following information on their commuter flights from Boston to New York:
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35. Rules Of Addition Example 1
Busn 210 Chapter 5
Rules Of Addition Example 1
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36. Rules Of Addition Example 2
Busn 210 Chapter 5
Rules Of Addition Example 2
In a sample of 500 students:
320 said they had a music sound system P(S)
175 said they had a TV P(TV)
100 said they had both P(S and TV)
5 said they had neither TV
175
Both
100 S
320
In this circumstance, Events P(S and TV) are not mutually exclusive!
What is the sample space?
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37. Rules Of Addition Example 2
Busn 210 Chapter 5
Rules Of Addition Example 2
If a student is selected at random, what is the probability that the student has:
Only a music sound system
Only a TV
Both a music sound system and TV
P(S) = 320/500 = .64
P(TV) = 175/500 = .35
P(S and TV) = 100/500 = .20
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38. Rules Of Addition Example 2
Busn 210 Chapter 5
Rules Of Addition Example 2
If a student is selected at random, what is the probability that the:
Student has only a music sound system or TV?
Student has both a music sound system and TV?
P(S or TV) = P(S) + P(TV) - P(S and TV)
= 320/ /500 – 100/500
= .79
P(S and TV) = 100/500
= .20
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39. Special Rule Of Multiplication
Busn 210 Chapter 5
Special Rule Of Multiplication
The special rule of multiplication requires that two events A and B are independent
Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other
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40. Special Rule Of Multiplication Example 1
Busn 210 Chapter 5
Special Rule Of Multiplication Example 1
Chris owns two stocks:
IBM
General Electric (GE)
The probability that IBM stock will increase in value next year is .5
The probability that GE stock will increase in value next year is .7
Assume the two stocks are independent
What is the probability that both stocks will increase in value next year?
P(IBM and GE) = (.5)(.7) = .35
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41. Special Rule Of Multiplication Example 2
Busn 210 Chapter 5
Special Rule Of Multiplication Example 2
If the probability of selecting a finished boomerang with a blemish in the paint job is .02, what is the probability of randomly selecting four boomerangs from the production line (boomerangs just rolling off the line) and finding all four blemished?
Because there are so many, we can assume independence
P(selecting 1 boom with blemish) = .02
P(selecting 4 booms with blemish) = .02*.02*.02*.02=
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42. General Multiplication Rule
Busn 210 Chapter 5
General Multiplication Rule
The general rule of multiplication is used to find the joint probability that two events will occur
It states that for two events A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of B given that A has occurred
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43. General Multiplication Rule Example 1
Busn 210 Chapter 5
General Multiplication Rule Example 1
Now you have ten boomerangs and two of them have blemishes
We want to select one after the other
What is the probability of selecting a blemished boom followed by another blemished boom?
The sample space will change (without replacement)
The second P(X) is dependent on the first
P(Bblemish1)* P(Bblemish2) = 2/10*1/9 = 2/90  .0222
Example 2:
In class example with women & men: what is the probability of selecting from a hat the names of three women
10/20*9/19*8/18 =
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44. Busn 210 Chapter 5
In Class
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45. Busn 210 Chapter 5
A contingency table is used to classify observations according to two identifiable characteristics.
Contingency tables are used when one or both variables are nominally or ordinally scaled.
A contingency table is a cross tabulation that simultaneously summarizes two variables of interest.
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46. Busn 210 Chapter 5
General Multiplication Rule Example 2 Contingency Table (Cross-classified)
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47. Busn 210 Chapter 5
General Multiplication Rule Example 2 Contingency Table (Cross-classified)
80/200
35/200
Addition Rule:
P(Would Not Remain or Has Less Than 1 Year Experience) =
80/ /200 – 25/200 = 90/200 = .45
P(Select 1-5 Years Experience) = 45/200
P(Would Not Remain given that 1-5 Years) = 15/45
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48. Use A Tree Diagram To Organize And Compute Probabilities
Busn 210 Chapter 5
Use A Tree Diagram To Organize And Compute Probabilities
Each segment of the tree is one stage in the problem
The branches of a tree diagram are weighted by probabilities
Steps:
Draw heavy dots on left to represent the root of the tree
Two main branches are drawn with “root probabilities”
Create branches for each conditional probability
Write out Joint Probabilities
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49. Busn 210 Chapter 5
Draw Heavy Dots On Left To Represent The Root Of The Tree & Draw Two Main Branches With “Root Probabilities”
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50. Create Branches For Each Conditional Probability
Busn 210 Chapter 5
Create Branches For Each Conditional Probability
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51. Write Out Joint Probabilities
Busn 210 Chapter 5
Write Out Joint Probabilities
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52. Don’t Forget To Extend The Rules:
Busn 210 Chapter 5
Don’t Forget To Extend The Rules:
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53. Some Principles Of Counting
Busn 210 Chapter 5
Some Principles Of Counting
Multiplication Formula
Combination Formula
Permutation Formula
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54. Multiplication Formula
Busn 210 Chapter 5
Multiplication Formula
The multiplication formula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both
m x n indicates the number of ways they can be done in sequence, or the number of outcomes, or number of arrangements
Example: Dr. Delong has 10 shirts and 8 ties. How many shirt and tie outfits does he have? (10)(8) = 80
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55. Busn 210 Chapter 5
n! = n Factorial
5! = 1*2*3*4*5 = 120
5!/3! = 1*2*3*4*5/1*2*3 = 4*5 = 20
Use your Calculator
Use Excel Functions (see page 169)
******Note
0! = 1
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56. Combinations Order Not Important
Busn 210 Chapter 5
Combinations Order Not Important
A combination is the number of ways to choose r objects from a group of n objects without regard to order
Note: The order of arrangement is not important in permutations
a, b, c is same as b, c, a
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57. Busn 210 Chapter 5
Combination Example
There are 12 players on the Highline basketball team. Coach Che Dawson must pick five players among the twelve on the team to comprise the starting lineup. How many different groups are possible?
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58. Permutations Order Important
Busn 210 Chapter 5
Permutations Order Important
A permutation is any arrangement of r objects selected from n possible objects
Note: The order of arrangement is important in permutations
a, b, c is not the same as b, c, a
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59. Busn 210 Chapter 5
Permutations Example
Suppose that in addition to selecting the group, Che Dawson must also rank each of the players in that starting lineup according to their ability
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60. Busn 210 Chapter 5
Law of Large Numbers
Suppose we toss a fair coin. The result of each toss is either a head or a tail. If we toss the coin a great number of times, the probability of the outcome of heads will approach .5. The following table reports the results of an experiment of flipping a fair coin 1, 10, 50, 100, 500, 1,000 and 10,000 times and then computing the relative frequency of heads
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