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Chapter 3: Probability

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Title: Chapter 3: Probability


1
Chapter 3Probability
  • The Cartoon Guide to Statistics
  • By Larry Gonick
  • As Reviewed by
  • Michelle Guzdek

GEOG 3000 Prof. Sutton
2
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.

3
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.

4
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.

5
Sample Space
For a single die
For a pair of dice
6
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

7
Probability Histogram
8
Probability Histogram
9
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
10
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

11
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.

12
Modeled Probability vs. Relative Frequency
13
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

14
Event Dice Add to 7
15
Calculate the events
16
Answer
17
Calculate Probability
18
Answer
19
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)

20
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)

21
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)

22
Bayes Theorem
23
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
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