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Probability

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					Random Phenomena
   Outcome is unknown before the event
     Flipping a coin
     Rolling a die
     Taking a sample from a population

   Long term behavior is predictable
     50% heads
     16.67% for 1, 2, 3, 4, 5, or 6
     Sampling Distribution (Chapter 18)


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Probability
   Subjective (Personal)
       Based on feeling or opinion.
   Empirical
       Based on experience.
   Theoretical (Formal)
       Based on assumptions.



                                       2
Chips Example
   Bag o‟ chips (poker chips).
       Some are red.
       Some are white.
       Some are blue.
   Draw a chip from the bag.
   We don‟t know how many of each of the three
    colors are in the bag.
   Assumption: Each chip has equal probability of
    being chosen (Same size, same weight, same
    texture, etc.)
                                                     3
The Deal
   Possible out comes of the draw

     Draw a blue chip win 3 bonus points.
     Draw a red chip win 2 bonus points.

     Draw a white chip lose 3 bonus points.




                                               4
Is this a good deal?
   Subjective (personal) probability
       Based on your beliefs and opinion.
   Empirical probability
     Based on experience.
     Conduct a series of trials.

     Each trial has an outcome (R, W, B).




                                             5
Empirical Probability
   Look at the long run relative frequency of
    each of the outcomes after choosing n=50
    with replacement.
     Blue
     Red

     White




                                             6
Theoretical Probability
   Look in the bag and see how many
     Blue chips –
     Red chips –

     White chips –

   Assumption
       Each chip has the same probability of being
        chosen. Equally likely.


                                                      7
Law of Large Numbers
   For repeated independent trials, the long
    run relative frequency of an outcome gets
    closer and closer to the true probability of
    the outcome.
       How does this compare with the “Law of
        Averages”?




                                                 8
Law of Large Numbers
   Probability is a long term number
       Ex. Flip a coin 5 times and get 5 heads in a
        row, is a tail due on next flip?
   Random events do not compensate for
    short term behavior
       Over a long sequence of flips, even after a
        sequence of many heads in a row,
        P(tails after sequence) = 0.5

                                                       9
 Law of Large Numbers
 Over the long term, P(heads) = 0.5
 Long term - Infinite




                                       10
Probability Rules
 A probability is a number between 0 and
  1.
 Something has to happen rule.
       The probability of the set of all possible
        outcomes of a trial must be 1.




                                                     11
Probability Rules
   Event – a collection of outcomes.
       Win bonus points (Blue or Red chip)
   Complement rule
     The probability an event occurs is 1 minus
      the probability that it doesn‟t occur.
     P(A) = 1 – P(AC)




                                                   12
Probability Rules
  Disjoint events – no outcomes in
   common.
  Addition Rule for disjoint events.
      P(A or B) = P(A) + P(B)
      P(Blue or Red) = P(Blue) + P(Red)




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Probability Rules
   Independent outcomes - The outcome of one
    trial does not influence the outcome of the
    other.
   1) Coin flips as „Head‟ and 2) Coin flips as
    „Head‟.
       INDEPENDENT: The outcome of flipping a coin does
        not depend on the previous coin flip outcome.
   1) It snows or not and 2) Class is cancelled or
    not
       NOT INDEPENDENT: The outcome of 2) depends
        on the outcome of 1).
                                                       14
Probability Rules
 Independent trials
 Multiplication rule for independent trials.
    P(1st outcome and 2nd outcome) =
     P(1stoutcome)*P(2nd outcome)




                                                15
Example
   What is the chance that two people in a
    row win bonus points?
       P(win 1st and win 2nd)=P(win 1st)*P(win 2nd)
       P(win 1st) = P(Blue or Red) = P(Blue)+P(Red)
       P(win 2st) = P(Blue or Red) = P(Blue)+P(Red)




                                                      16
Three Terms to Look For.
   Not
       This means the compliment that is subtract
        form 1.
   Or
       This means to add the probabilities.
   And
       This means to multiply the probabilities.


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