PROBABILITY

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• PROBABILITY

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Experiments WithUncertain Outcomes

• Experiment
• Toss a coin
• Roll a die
• Inspect a part
• Conduct a survey
• Hire New Employees
• Find Errors on Tax Form
• Complete a Task
• Weigh a Container

• Outcomes
• Heads/Tails
• 1, 2, 3, 4, 5, 6
• Defective/OK
• Yes/No
• 0, 1, 2 , 3
• 0 - 64
• 0-10 days
• 0 25 pounds

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Simple Events and Events

• Simple Event
• One of the possible outcomes (that cannot be
  further broken down)
• Sample Space
• Set of all possible simple events
• Mutually Exclusive
• Exhaustive
• Event
• A collection of one or more simple events

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PROBABILITY CONCEPTS

• Probability
• The likelihood an event will occur
• Basic Requirements for Assigning Probabilities
• The probability of all events lies between 0 and
  1
• The sum of the probabilities of all simple events
  1

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3 Approaches to Assigning Probabilities

• A priori Classical Approach
• Games of chance
• Relative Frequency Approach
• Long run likelihood of an event occurring
• Subjective Approach
• Best estimates

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Classical Approach

• Assume there are N possible outcomes of an
  experiment and they are all equally likely to
  occur
• Assigning Probability
• Suppose X of the outcomes correspond to the event
  A. Then the probability that event A will occur,
  written P(A) is
• P(A) X/N
• Example P(Club) clubs/52 13/52

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Relative Frequency Approach

• Long term behavior of an event A has been
  observed
• n observations
• P(A) (times A occurred) / n
• Example n 800 students take statistics
• 164 received an A
• P(Receiving an A) 164/800

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Subjective Approach

• These are best estimate probabilities based on
  experience and knowledge of the subject
• Example A meteorologist uses charts of wind
  flow and pressure patterns to predict that the
  P(it will rain tomorrow ) .75
• This will be stated as a 75 chance of rain
  tomorrow

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PROBABILITIES OF COMBINATIONS OF EVENTS

• Joint Probability
• P(A and B) Probability A and B will occur
  simultaneously
• Marginal Probability
• P(A) ?(Probabilities of all the simple events
  that contain A)
• Either/Or Probability -- Addition Rule
• P(A or B) P(A) P(B) - P(A and B)
• Conditional Probability
• P(AB) P(A and B)/P(B)
• Joint Probability (Revisited)
• P(A and B) P(AB)P(B) P(BA)P(A)
• Complement Probability

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INDEPENDENCE

• Events A and B are independent if knowing B does
  not affect the probability that A occurs or vice
  versa, i.e.
• P(AB) P(A) and P(BA) P(B)
• Joint Probability (For Independent Events)
• P(A and B) P(A)P(BA) P(A)P(B) if A and B
  are independent
• A Test for Independence -- Check to see if
• P(A and B) P(A)P(B)
• If it does gt Independent If not,
  gt Dependent

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Mutually Exclusive and Exhaustive Events

• Events A and B are mutually exclusive if
• P(A and B) 0
• Thus if A and B are mutually exclusive,
• P(A or B) P(A) P(B) - P(A and B)
  P(A) P(B)
• Events A, B, C, D are exhaustive if
• P(at least one of these occurs) 1

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Example

• 200 people from LA, OC and SD surveyed
• Do you favor gun control?
• YES NO ?
• LA 40 30 10
• OC 50 10 20
• SD 10 30 0

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Joint Probability Table

• Example
• P(LA) P(LA and Yes) P(LA and NO) P(LA and
  ?)
• .20 .15 .05
  .40

Marginal .40 P(LA) .40 P(OC) .20 P(SD)
YES NO ? LA
.20 .15 .05 OC .25
.05 .10 SD .05 .15 0
Marginal P(YES) P(NO) P(?)
.50 .35 .15
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What is the probability a randomly selected
person is from LA and favors gun control?
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What is the probability a randomly selected
person is opposed to gun control?
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What is the probability a randomly selected
person is not from San Diego?
.80
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Joe is from LA. What is the probability Joe
favors gun control?
We know (we are given that) Joe is from LA.
.50
P(YESLA) P(YES and LA)/P(LA)
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Bill is opposed to gun control. What is the
probability Bill is from Orange County?
We know (we are given that) Bill is opposed to
gun control.
P(OCNO) P(OC and NO)/P(NO)
.143
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What is the probability that a randomly selected
person is from LA or favors gun control?
P(LA or Yes) P(LA) P(YES) - P(LA and YES)
.70
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Are being from San Diego and having no opinion on
gun control a pair of mutually exclusive events?
Does P(SD and ?) 0
They are mutually exclusive.
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Are being from Orange County and having no
opinion on gun control a pair of mutually
exclusive events?
Does P(OC and ?) 0
They are not mutually exclusive.
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Are being from LA and favoring gun control a pair
of independent events?
LA and YES are independent
Does P(LA and YES) P(LA)P(YES)?
.20
YES
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Are being from San Diego and favoring gun control
a pair of independent events?
SD and YES are not independent
Does P(SD and YES) P(SD)P(YES)?
.10
NO
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Are being from LA, being from Orange County,
favoring gun control, and opposing gun control
form a set of exhaustive events?
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Are being from Orange County, being from San
Diego, favoring gun control, and opposing gun
control form a set of exhaustive events?
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Calculating Probabilities Using Venn Diagrams

• Convenient way of depicting some of the logical
  relationships between events
• Circles can be used to represent events
• Overlapping circles imply joint events
• Circles which do not overlap represent mutually
  exclusive events
• The area outside a region is the complement of
  the event represented by the region

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Example

• Students at a college have either Microsoft
  Explorer (E), Netscape (N), both or neither
  browsers installed on their home computers
• P(E) .85 and P(N) .50 P(both) .45
• What is the probability a student has neither?

P(E or N) .85.50-.45 .90
P(neither) 1-.90 .10
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Probability Trees

• Probability Trees are a convenient way of
  representing compound events based on conditional
  probabilities
• They express the probabilities of a chronological
  sequence of events
• Example
• The probability of winning a contract is .7.
• If you win the contract P(hiring new workers)
  .8
• If you do not win the contract P(hiring new
  workers) .4
• What is the probability you will hire new
  workers?

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The Probability Tree

• Start with whether or not you win the contract
• Then for each possibility list the probability of
  hiring new workers
• Multiply the probabilities and add appropriate
  ones

.56
.14
.12
.18
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REVIEW

• Probabilities are measures of likelihood
• How to determine probabilities
• Joint, marginal, conditional probabilities
• Complement and either/or probabilities
• Mutually exclusive, independent and exhaustive
  events
• Venn diagrams
• Decision trees