Probability

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					Probability

  Topic 4
                    Probability
• Until now we‟ve discussed descriptive statistics
   – Methods for organizing and summarizing data
   – Graphical and numerical displays
   – Linear regressions
• Collection and production of data
   –   Sample design
   –   Observational studies
   –   Design of Experiments
   –   Simulation
                Probability
• We will now study a few more topics that
  will set the stage for our study of inferential
  statistics
• We‟ve mentioned already that inferential
  statistics involves making conclusions about
  a population based on information/data
  collected from a sample from that
  population
                 Probability
• Because we are drawing conclusions about the
  population based on a sample, we can never be
  certain that our conclusions are 100% correct.
• That is, there is some uncertainty.
• Probability is the study of uncertainty and
  provides some of the foundation for our study of
  inferential statistics – confidence intervals and
  hypothesis testing.
      Spinning Wheel Activity
• This experiment consists of spinning the
  spinner 3 times and recording the numbers
  as they occur. We want to determine the
  proportion of times that at least one digit
  occurs in the correct position. For example,
  in the number 123, all of the digits are in
  their proper positions, but in the number
  331, none are.
    Guess, Experiment, Theory
• First, let‟s guess the proportion of times at
  least one digit will occur in the proper
  position.
• How would you simulate this if you didn‟t
  have a spinner?
• Let‟s do the activity.
• Later we‟ll calculate the theoretical
  probability.
                   Probability
•   Chance exists all around us.
•   Human design – casinos
•   In nature – sex of a child
•   Probability is the study of chance or uncertainty.
•   Chance behavior – like our activity – that seems
    haphazard and unpredictable in the short run, has a
    regular and predictable pattern the the long run.
      Let‟s do another activity
• Flipping a coin
• Let‟s flip it 100 times and record/graph the
  proportion of heads.
• In the short run, this proportion or
  probability is unpredictable.
• In the long run, the proportion of heads
  approaches the probability of getting ahead
  -- .5.
          Random vs Haphazard
• Random is not synonymous with haphazard.
• Random is a description of a kind of order that
  emerges in the long run.
   – Uncertain but a regular pattern in the long run
• We often encounter the unpredictable side of
  randomness in everyday life, but we rarely see
  enough repetitions of the same phenomenon to
  observe the long run regularity that probability
  describes.
• Probability is the proportion of times an outcome
  occurs in the long run – long term relative
  frequency.
• Probability theory is the study of uncertainty, the
  study of random behavior.
                 Terminology
• Chance experiment: an activity or situation where
  there is uncertainty about what outcome out of
  many possible outcomes will occur.
   – Our spinning wheel activity
   – Equal Likelihood model
• Sample space: the collection of all possible
  outcomes of a chance experiment
   – {123} {121} {122} {111} …
   – Tree diagram
           More Terminology
• Event: collection of outcomes from the
  sample space of a chance experiment
  – A: second digit is correct
• Simple event: an event that is made up of
  exactly one outcome
  – B: all digits are correct
   Notation and Displaying Events

• Notation: E = event all digits in correct place
• Venn Diagrams




                      E

                            not E
    Relationship Among Events
• Complement
   – AC
   – The event A does not occur
   – Venn Diagram
• Intersection               

   – A  B „A and B’
   – The event A and B occur
   – Venn Diagram
• Union
   – A B „A or B’
   – The event either A or B occurs
                    Examples
• Playing Cards
  –   A = card selected is a kings of hearts
  –   B = card selected is a king
  –   C = card selected is a heart
  –   D = card selected is a face card
  –   a. (not D) b. (B and C) c. (B or C)
• Describe the events
       More about relationships
• Mutually exclusive or disjoint events: two or
  more events are said to be mutually exclusive if at
  most one of them can occur when the experiment
  is performed, that is, if no two events have
  outcomes in common
• E = card selected is black
• F = card selected is a diamond
• D = card selected is a face card
• Relationship between E and F, E and D, F and D?
      Approaches to Probability
• Probability is the study of uncertainty – the
  mathematical characterization of uncertainty
• Notation: P(A) =
• Interpretation of Probability:
   – Near 0 – unlikely
   – Near 1 – very likely
   – Frequentist interpretation – proportion of times an event
     will occur in a large repetition of trial of the experiment
• The Classical Approach
• The Relative Frequency Approach (Empirical
  Approach)
• The Subjective Approach
          Subjective Approach
• We‟ve already seen this approach during the
  sinning wheel activity when I asked you what you
  thought the proportion for digits in correct places
  would be
• We use the subjective approach to quantify
  likelihood of events all the time
   – Should I wear a rain coat
   – Should I speed
   – Should I do my homework
          Empirical Approach
• Also called the relative frequency approach
• Using an experiment or a simulation allows us to
  determine probabilities using the empirical
  approach
•
     P E  
              number of times E occurs
                total number of trials
• As the number of repetitions of a chance
  experiment increases, the chance that the relative
  frequency of occurrence for an event will differ
  from the true probability of that event by more
  than a small amount approaches 0
       Law of Large Numbers
• As our number of trials grows larger and
  larger, the relative frequency of our event
  approaches the true probability of that
  event.
    Examples of the Empirical
           Approach
• Spinner Activity
• Flipping the coin
• For equally likely outcomes, you can arrive
  at the correct probability using either
  approach
• Not so with outcomes that are not equally
  likely
           Classical Approach
• First, some notation – probability of an event
• Classical Approach says
  P E  
             number of outcomes favorable to E
           number of outcomes in the sample space
• To use the classical approach to determine
  probabilities, all outcomes must be equally likely
  – games of chance
• Go back to our spinner activity to calculate the
  probability
     Limitation to the Classical
             Approach
• Consider accident rates of young adults driving
  cars. Who do you think has a higher insurance
  rate, an 18 year old or a 35 year old? Why?
• Using the classical approach when you consider
  an insurance policy for an 18 and 35 year old,
  there are 2 outcomes –having an accident and not
  having an accident – and those outcomes are
  equally likely for the 18 and 35 yo. The
  probabilities that each would have an accident
  using the classical approach is .5. Which defies
  reason and experience.
  Basic Properties of Probability
• For any event E, 0  P( E )  1
• If S is the sample space for an experiment,
  then P(S )  1
• If two events E and F are mutually
  exclusive, then P( E or F )  P( E )  P( F )
   – General Addition Rule: P( AorB)  P( A)  P( B)  P( A & B)
• For any event E, P( E )  P( E C )  1
            Conditional Probability
• Sometimes the knowledge that an event occurred
  changes the likelihood that another event will
  occur
• A simple example. Let‟s say we have 2 events
    –   A: You choose a face card
    –   B: You choose a King
    –   Without any other info, what is P(B)?
    –   What is the probability of choosing a King given you have chosen a
        face card?
• Let‟s say 1% of the population has a certain disease. You can‟t tell if
  you have the disease unless you are tested. Let‟s say that 80% of the
  people who test positive have the disease – 20% who test positive
  don‟t have the disease.
    –   E: You have the disease
    –   F: You test positive
    –   Without any info, what is P(E)?
    –   Let’s say you test positive, what is P(E)? In other words, what is the
        probability you have the disease given you test positive?
      Notation for Conditional
            Probability
•   P( E \ F )

• Read as “the probability of event E given F
  has occurred”
 An example using two-way tables
• The ASU Statistical Summary provides information on
  various characteristics of the ASU faculty. Data on age
  and rank of ASU faculty is presented in the table below.

               Full   Assoc   Asst   Inst    Total
               Prof   Prof    Prof
       <30     2      3       57     6       68

       30-39   52     170     163    17      402

       40-49   156    125     61     6       348

       50-59   145    68      36     4       253

       >60     75     15      3      0       93

       Total   430    381     320    33      1164
    Determine some probabilities
•   Let‟s select faculty at random
•   P(Asst Prof)
•   P(30-39)
•   P(Full Prof \ 40-49)
•   P(50-59 \ Asst Prof)
                 Definition
             P( E  F )
• P(E \ F) =
               P( F )

• Let‟s use this definition to determine
  – P(Instr \ 30-39)
  – P(40-49 \ Asst Prof)
                 A Summary
•   Probability – the study of uncertainty
•   Chance experiment
•   Event/Simple event/Sample space
•   Venn Diagrams – forming new events
•   Complement, union, intersection, disjoint
•   Determining probabilities
    – Classical Approach
    – Empirical Approach
• Law of Large Numbers
• Properties of Probability
• Conditional Probability
               Independence
• In conditional probability, the knowledge of one
  having occurred affects the likelihood that some
  other event will occur.
• It is also possible that the knowledge of one event
  occurring will not change the probability of
  occurrence of a second event – these events are
  independent.
• Can you think of a simple chance experiment
  where events are independent?
  An example from the text
            Single     Condo     Multi     Total
            Family      (C)     Family
              (F)
   ARM         .4       .21       .09        .7
    (E)
   Fixed      .10       .09       .11        .3


   Total       .5        .3        .2


Determine the following probabilities:
        Mort is ARM given mort is for single family home
        Mort is for single family home given is an ARM
        Mort is ARM given mort is for Condo
        Mort is an ARM
                 Definition
• Two events E and F are said to be
  independent if
          P( E \ F )  P( E )
                     P( E \ F )  P( E )

• If E and F are not independent, they are call
  dependent events
• If P( E \ F )  P( E ) then P( F \ E )  P( F )
• Nothing we learn about one event will
  change the likelihood of the other event.
       Multiplication Rule for
        Independent Events
• If events E and F are independent, then

      P( E  F )  P( E )  P( F )
     Sampling with and without
           replacement
• We‟ve talked about sampling in earlier topics. We
  made sure our samples were random – basically,
  the probability of being selected is the same for
  each individual and each sample of size n.
• When we sample with replacement, thisis easy to
  accomplish.
• In reality, when we sample we sample without
  replacement – but this introduces bias doesn‟t it?
• Under certain circumstances, selections in
  sampling without replacement -- independence is
  not a cause for concern
      Relatively small sample
• If a random sample of size n is taken from a
  population of size N, independence can be
  assumed when N is at least 20 times larger
  than n. The theoretical probabilities of
  selecting without replacement differ
  insignificantly from the theoretical
  probabilities of selecting with replacement.
• Examples 6.20 and 6.21
         HW I (Indpendence)
• Read POD Section 6.5
• Problems 6.36, 6.37, 6.38, 6.43, 6.44, 6.51,
  6.52

				
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posted:9/6/2011
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