Introducing Probability

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					Introducing Probability
                  Analogy
taking a simple random sample from a
population and calculating a summary statistic
analogous to game of chance:
     1. do random procedure with many
        possible outcomes
     2. end up with one particular outcome
     3. distribution of outcomes for large
        number of plays can be characterized
              Example
1. roll a fair die; numbers 1-6 are possible
    outcomes
2. we get, say, 4
3. if die fair, 1/6th of large number of
    rolls should give 1, 1/6th should give 2,
    etc.
iClicker
Consider flipping 2 coins and then counting the
number of heads. What are the possible
outcomes for this experiment?


 a.   1, 2
 b.   0, 1
 c.   0, 1, 2
 d.   1, 2, 3
iClicker
What do you think the long run proportion of
times you would get 1 head is?


 a.   1/2
 b.   1/3
 c.   1/4
 d.   2/3
           Probability Theory
components:
   1. specify game (including strategy if
      applicable)
   2. specify possible outcomes
   3. specify probability distribution = long
      run proportion associated with each
      possible outcome
            Probability Theory
theory can guide decision on strategy for
playing game:
     strategy that has higher probability (long-
     term proportion) of favorable results can
     be considered better strategy (even in
     short run)
      ‘Let’s Make a Deal’ game
1. 3 doors
2. car placed randomly behind one door
   (other 2 doors have goats )
3. you choose a door
4. non-chosen door with goat shown to you
5. given option to switch doors
     • strategy 1: don’t switch
     • strategy 2: switch
6. chosen door is opened – win car or goat
       ‘Let’s Make a Deal’ game
•  possible outcomes: goat, car
•  probability distribution of outcomes
   depends on strategy
• can find probability distribution in 2 ways:
  – theoretical calculations
  – empirical evaluation: actually play the
     game (or simulate it) thousands of times
  – often counterintuitive (don’t trust
     intuition)
iClicker
Which strategy do you think results in the
highest probability of getting a car?



a. switch
b. don’t switch
c. both are the same
               Terminology
Random phenomenon
    • individual outcome unpredictable, but
      outcomes from large number of
      repetitions follow regular distribution
Sample space
    • set of all possible outcomes
              Terminology
Probability of an outcome
    • proportion of times the outcome would
      occur in large number of repetitions
    • number between 0 and 1
                Terminology
Probability distribution
     • set of probabilities for all outcomes in
       sample space
     • must sum to 1
     • can be represented by table, formula,
       histogram-like graph
     • if phenomenon simulated thousands of
       times, histogram of results looks like
       probability distribution of phenomenon
 Random Sampling or Randomized
          Experiment
true random phenomenon with statistic as
  outcome
    • knowledge of nature and spread of
      probability distribution of statistic allows
      assessment of uncertainty associated with
      particular observed statistic
iClicker
Consider taking a simple random sample of 100
BYU students, asking them whether they ski,
snowboard, or neither, and then calculating the
percent that answer ‘snowboard’. What is the
sample space?

a. all BYU students
b. 100 BYU students
c. ski, snowboard,
   neither
d. 0% - 100%
              Vocabulary
Long Run Proportion
Outcome
Probability
Probability Distribution
Probability Model
Random Phenomenon
Statistic
Simulation
Simple Random Sampling

				
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posted:11/25/2012
language:English
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