Chapter 3: Probability

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Chapter 3Probability

• The Cartoon Guide to Statistics
• By Larry Gonick
• As Reviewed by
• Michelle Guzdek

GEOG 3000 Prof. Sutton
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It all started with gambling

• No one knows when it started, but it at least
  goes as far back as Ancient Egypt.
• The Roman Emperor Cladius
• (10BC 54 AD) wrote the first
• book on gambling.
• Dice grew popular in the
• Middle Ages.

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Basic Definitions

• Random Experiment the process of observing the
  outcome of a chance event.
• Elementary Outcome all possible results of the
  random experiment.
• Sample Space the set or collection of all the
  elementary outcomes.

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Coin Toss Example

• The random experiment consists of recording the
  outcome.
• The elementary outcomes are heads and tails.
• The sample space is
• the set written as H,T.

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Sample Space
For a single die
For a pair of dice
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Probability

• Probability is a numerical weight assigned to a
  possible outcome.
• In a fair game of heads and tails, the outcomes
  are equally likely so probability is .5 for both.
• P(H) P(T) .5
• For two dice, there are 36 elementary outcomes
  all equally likely.
• P(BLACK 5, WHITE 2) (1/6)(1/6) 1/36

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Probability Histogram
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Probability Histogram
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What if

• What if things were not equal and a gambler
  throws loaded die?
• Now P(1) .25, and the remaining probabilities
  must equal 1 - .25 .75.
• It 2,3,4,5, and 6 are equally likely to occur the
  probability of each is
• .75/5 .15

P(x) .25 .15 .15 .15 .15 .15
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Random Experiment

• Probabilities are never zero.
• A probability of zero means it cannot happen.
• Less than zero would be meaningless.
• Therefore
• P(Oi) 0
• If an event is certain to happen we assign
  probability of 1.
• Combine these two and you have the Characteristic
  Properties of Probability
• P(Oi) 0
• P(O1) P(O2) P(On) 1

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Approaches to Probability

• Classical Probability
• Based on gambling ideas.
• Assumption is the game is fair and all elementary
  outcomes have the same probability.
• Relative Frequency
• When an experiment can be repeated, then an
  events probability is the proportion of times
  the event occurs in the long run.
• Personal (Subjective) Probability
• Lifes events are not repeatable.
• An individuals personal assessment of an
  outcomes likelihood. For example, betting on a
  horse.

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Modeled Probability vs. Relative Frequency
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Basic Operations

• An EVENT is a set of elementary outcomes.
• The probability of an event is the sum of the
  probabilities of the elementary outcomes in the
  set.
• You can combine events to make other events,
  using logical operations.
• AND, OR or NOT

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Event Dice Add to 7
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Calculate the events
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Answer
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Calculate Probability
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Answer
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Addition Rule

• Mutually exclusive not overlap
• P(E OR F) P(E) P(F)
• Overlap of elementary outcomes
• P(E OR F) P(E) P(F) P(E AND F)
• When P(NOT E) is easier to compute use
  subtraction rule.
• P(E) 1 P(NOT E)

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Conditional Probability

• The probability of A, given C
• P(AC) P(E AND F)/P(F)
• When E and F are mutually exclusive
• P(EF) 0, once F has occurred E is impossible
• Rearranging the definition get multiplication
  rule
• P(E AND F) P(EF)P(F)

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Independence

• Two events E and F are independent of each other
  if the occurrence of one had no influence on the
  probability of the other.
• P(E AND F) P(E)P(F)

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Bayes Theorem
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References

• DiFranco, Steven. Chapter 3 Probability
  Distributions, 2009. http//www.difranco.net/qmb2
  100/Lecture_notes/Chapter_3/ch03hb.htm
• Hajek, Alan. Interpretations of Probability,
  2009.
• http//plato.stanford.edu/entries/probability-in
  terpret/
• Joyce, James. Bayes Theorem, 2003.
  http//plato.stanford.edu/entries/bayes-theorem/
  
• Khan Academy (YouTube Username khanacademy).
  Probability (part 1), 2008. http//www.youtube.co
  m/watch?v3ER8OkqBdpE
• Khan Academy (YouTube Username khanacademy).
  Probability (part 5), 2008. http//www.youtube.co
  m/watch?v2XToWi9j0Tk
• Spaniel, William (YouTube Username
  JimBobJenkins). Game Theory 101 Basic
  Probability Rules, 2009. http//www.youtube.com/w
  atch?vdFSWW6QTVp0
• Waner, Stefan and Steven Constanoble. 7.3
  Probability and Probability Models, 2009.
  http//www.zweigmedia.com/RealWorld/tutorialsf15e
  /frames7_3.html
• Interactive quizzes