A Survey of Probability Concepts

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Chapter 5

• A Survey of Probability Concepts

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Goals

• Probability Concepts
• Define probability
• Understand the terms
• Experiment
• Outcome
• Event
• Describe these approaches to probability
• Classical
• Empirical
• Subjective

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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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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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Statistical Inference OrInferential 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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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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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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• 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?

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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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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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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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Define Event

• When one or more of the experiments 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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Experiment, Outcome, Event
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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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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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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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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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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(BA) 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 dont 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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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(BA)
• 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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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

• Root probabilities times conditional
  probabilities equal joint probabilities (Tree
  Diagrams)

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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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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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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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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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Empirical Approach To Probability

• Throughout her teaching career Professor Jones
  has awarded 186 As 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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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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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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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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In this circumstance, Events A B are not
mutually exclusive!
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Calculate Probabilities Applying These Rules

• Rules of addition
• Rules of multiplication

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Rules Of Addition
Why? Dont want to count twice! P(X) ? 1
HW 22, page 152 at least one either or
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Rules Of Addition
Mutually Exclusive!
Event B P(B)
Event A P(A)
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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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Rules Of Addition Example 1
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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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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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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)
320/500 175/500 100/500 .79 P(S and TV)
100/500 .20
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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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Special Rule Of MultiplicationExample 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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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)
  .02.02.02.02.000000160

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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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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) 2/101/9 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
• 10/209/198/18 .105263158

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In Class
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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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General Multiplication Rule Example 2Contingency
Table (Cross-classified)
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General Multiplication Rule Example 2Contingency
Table (Cross-classified)
80/200
35/200
Addition Rule P(Would Not Remain or Has Less
Than 1 Year Experience) 80/200 35/200
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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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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Draw Heavy Dots On Left To Represent The Root Of
The Tree Draw Two Main Branches With Root
Probabilities
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Create Branches For Each Conditional Probability
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Write Out Joint Probabilities
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Dont Forget To Extend The Rules
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Some Principles Of Counting

• Multiplication Formula
• Combination Formula
• Permutation Formula

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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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n! n Factorial

• 5! 12345 120
• 5!/3! 12345/123 45 20
• Use your Calculator
• Use Excel Functions (see page 169)
• Note
• 0! 1

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CombinationsOrder 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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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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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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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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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