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Chapter 3Probability

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

GEOG 3000 Prof. Sutton

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.

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.

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.

Sample Space

For a single die

For a pair of dice

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

Probability Histogram

Probability Histogram

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

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

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.

Modeled Probability vs. Relative Frequency

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

Event Dice Add to 7

Calculate the events

Answer

Calculate Probability

Answer

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)

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)

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)

Bayes Theorem

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