Transcript Chapter 2: Fundamental Research Concepts

Chapter 3:
Probability
The Cartoon Guide to Statistics
By Larry Gonick
As Reviewed by:
Michelle Guzdek
GEOG 3000
Prof. Sutton
1/24/2010
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?
P(x) = .25


.15
.15
.15
.15
.15
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
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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
event’s probability is the proportion of times the event
occurs in the long run.
Personal (Subjective) Probability
 Life’s events are not repeatable.
 An individual’s personal assessment of an outcome’s
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


Overlap of elementary outcomes


P(E OR F) = P(E) + P(F)
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”


When E and F are mutually exclusive


P(A|C) = P(E AND F)/P(F)
P(E|F) = 0, once F has occurred E is
impossible
Rearranging the definition get
multiplication rule

P(E AND F) = P(E|F)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/qmb2100/Lecture_notes/Chapter_3/ch03hb.htm

Hajek, Alan. Interpretations of Probability, 2009.
http://plato.stanford.edu/entries/probability-interpret/

Joyce, James. Bayes’ Theorem, 2003.
http://plato.stanford.edu/entries/bayes-theorem/

Khan Academy (YouTube Username: khanacademy). Probability (part
1), 2008. http://www.youtube.com/watch?v=3ER8OkqBdpE
Khan Academy (YouTube Username: khanacademy). Probability (part
5), 2008. http://www.youtube.com/watch?v=2XToWi9j0Tk
Spaniel, William (YouTube Username: JimBobJenkins). Game Theory
101: Basic Probability Rules, 2009.
http://www.youtube.com/watch?v=dFSWW6QTVp0
Waner, Stefan and Steven Constanoble. 7.3: Probability and
Probability Models, 2009.


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http://www.zweigmedia.com/RealWorld/tutorialsf15e/frames7_3.html
•
Interactive quizzes
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