chance and probability 1 krishna.v.palem kenneth and audrey kennedy professor of computing...
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Chance and Probability
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Krishna.V.PalemKenneth and Audrey Kennedy Professor of ComputingDepartment of Computer Science, Rice University
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ContentsNumbers and History of Chance Relative Frequency and In Class ExperimentEvents and Definition of ProbabilityExercisesCourse Information
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ContentsNumbers and History of Chance Relative Frequency and In Class ExperimentEvents and Definition of ProbabilityExercisesCourse Information
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Cause and Effect of numbers
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The reason numbers were invented is to quantify things that are happening around us.Like the motion of planets.
The effect of numbers were manifold.The trade and finance industry is built on numbers.Science is based on numbers.The multi-billion gambling industry is built on
numbers.
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Time and Planetary motion
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One of the most important contributions of numbers was the ability to measure time
Time has always been measured in terms of the interval for the heavenly bodies to completeone cycle.
Numbers could now quantifythe precise amount of time takenfor one cycle.
In fact Indian mathematicians and astronomers(500AD) had calculated the diameter of the earth, length of the orbit of the earth etc. to an accuracy of 1%.
Only numbers led Nicolaus Copernicus to formulate a scientifically developed heliocentric cosmology.
It also then led to the Newton’s and Kepler’s laws that now are the basis of cosmology
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The multi-billion gambling industry
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The entire gambling industry is based on numbers.
Slot machines are designed in such a way that the casino always isat an advantage.
This is done by calculatingthe average chance of winningthat does not cause the casinoto lose.
Thus numbers provided a way of precisely quantifying chance.
Number is the basic tool to quantify and explain “Chance”
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Brief History of Chance
However, chance/gambling impulse predates humanity.
For centuries human beings speculated about probabilities in connection with • legal questions of evidence and contracts • insurance schemes
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The advent of ‘numbers’ provided people an opportunity to quantify chance.
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A 2005 Duke University study found that macaque monkeys preferred to follow a ‘riskier’ target, which gave them varying amounts of juice than the “safe” one, which always gave the same
Chance in Ancient World
Variety of animals, from bees to primates, embrace risk for a chance at a reward
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Chance in the Ancient World
On any given day, one might find lunch or become lunch.
Some events
just happen
Some can be influenced by seeking the
help of unseen spirits
Some can be
influenced
Classification of Events in Pre-History
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The hunter-gatherer lifestyle of early cultures was predicated on risk and reward.
like mining and fishing today
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Chance in Ancient WorldAs they discovered new technologies, humans gained
more control over their environment• but they still retained a fascination with chance.
• The emergence of more organized thought and language started to change essential ideas about events• developed the idea of ‘cause’ and ‘effect’
But whether every event has a cause in this world was a question not answered
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Explaining ChanceWhat do we mean when we say that
There is 50% chance to rain today.
It is 75% likely that the Houston Astros will win the game.
This means that these events can be predicted with a certain sense of certainty.
We will try to precisely define what this means.
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The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern- Keith Devlin
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The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World
Modern- Keith Devlin
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ContentsNumbers and History of Chance Relative Frequency and In Class ExperimentDefinition of ProbabilityExercisesCourse Information
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Events and Outcomes
Consider an experimentwhere one dice is rolled
Can show 6 differentvalues on the top side
Top side
1
2
3
4
5
6Top side
1
2
3
4
5
6
Hence this experiment is said tohave 6 different possible events
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Events and Outcomes (contd.)
Consider the same experimentwhere one dice is rolled
Top side
1
2
3
4
5
6
The outcome of the experiment is the actual result of performing the experiment
Let an experiment be to check if the value rolled is an even number
Then the favorable outcome in this experiment is the collection ofthe events {2,4,6}
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Mini-exercise 1 – Snakes and Ladders(S&L)Calculate the relative frequency of landing
in square 6 in the following scaled-down version of snakes and ladders (no snake or ladder)?
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7 8 9
6 5 4
1 2 3
The game is continues past 9.
What that means is that once you reach or cross 9 you come back to 1
So if you are at square 8 and roll a 3, then you come back to square 2
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Solution : Hint
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We will play 5 games The games would have 5, 10, 15 , 20 and 25 die rolls respectively
Record the square and compute the relative frequency for each game
Die roll Square landed
Favorable Outcome (6 )
Relative frequency =Number of favorable outcomes
Total number of outcomes
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ContentsNumbers and History of Chance Relative Frequency and In Class ExperimentDefinition of ProbabilityExercisesCourse Information
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Probability
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The classical definition of probability Pierre Simon
LaplaceThe probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.Let us elaborate…
Consider an event space
Event 1
Event 2
Event 3
Event 4
If there is no reasonbelieve that one event is more likely to occur
than another
Probabilityof favorableevents
Favorable events
No. of favorableevents
Total no. of events
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FairnessIn the snakes and ladders exercise
Was the probability of landing on all the squares equal ?
How many die rolls did you have to play until you reached that estimate ?Can you conclude that in 5 rolls ?
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7 8 9
6 5 4
1 2 3
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Generalizing Probability
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Consider another event space
Event 1
Event 2
Event 3
Event 4
Let us assume that each event is differently likelyto occur
Let us represent as a list of magnitude of“likeliness”
EventLikelines
s
Event 1 p1
Event 2 p2
Event 3 p3
Event 4 p4
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Generalizing Probability (Contd.)
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EventLikelines
s
Event 1 p1
Event 2 p2
Event 3 p3
Event 4 p4
Let us look at these values.
If these values have the following properties
1. All pi are in the range [0,1] 2. The sum of all pi = 1.
then these values are called the probabilities of theseevents.
For example, Event Probabiliti
es
Event 1 0.25
Event 2 0.25
Event 3 0.25
Event 4 0.25
Satisfies both the conditions
This means that event 1 occurs 1 out of 4 times … etc.
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Specification
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Consider the event space
Event 1
Event 2
Event 3
Event 4
Event Probabilities
Event 1 p1
Event 2 p2
Event 3 p3
Event 4 p4
Complete specification of the
behavior of the experiment
Both these together
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Now we have this relationship
Event
Event 1
Event 2
Event 3
Event 4
Probabilities
p1
p2
p3
p4
This can be defined concisely as a function whose independent variablerepresents the event
The dependent variable is the valueof the probability
Let the variable ‘x’ represent the event
x Event
1Event
1
2Event
2
3Event
3
4Event
4
ipixp )(Probability (Event i)
Here ‘x’ is called a random variable.Where i={1,2,3,4}
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Probability of more than one event
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RecallConsider an event space
Event 1
Event 2
Event 3
Event 4
Favorable events
Probabilityof favorableevents
No. of favorableevents
Total no. of events
Event Probabilities
Event 1 p1
Event 2 p2
Event 3 p3
Event 4 p4
Given this
Probabilityof favorableevents
21 pp
This is the probability of event 1 OR event 2
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ContentsNumbers and History of Chance Relative Frequency and In Class ExperimentDefinition of ProbabilityExercisesCourse Information
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Mini-exercise 2 – Snakes and Ladders(S&L)Calculate the probability of landing in
square 6 or square 8 in the following scaled-down version of snakes and ladders (no snake or ladder)?
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7 8 9
6 5 4
1 2 3
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Solution : Hint
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We will play 5 games The games would have 5, 10, 15 , 20 and 25 die rolls respectively
Record the square and compute the relative frequency for each game
Die roll Square landed
Favorable Outcome (6 or 8)
Relative frequency =Number of favorable outcomes
Total number of outcomes
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Building a modelAn important component of this course
(including exercises and projects) would be building models, and these games are an example.
An example of a model would be
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4
5
6
1/61/6
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Solution:Probability of reaching 6 via 5 = 1/6Probability of reaching 6 via 4 = 1/6
Total probability of reaching 6 from 4 OR 5 = 1/6 + 1/6 = 1/3
4
5
6
1/61/6
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This is the complete model of the scaled down snakes and ladder game
4
5
6
1/6
1/6
1/6 3
21
1/6
1/6
1/6
1/6
1/6
1/6
1/6
1/6
1/6
1/61/6
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4
5
6
1/6
1/6
1/6 3
21
1/6
1/6
1/6
1/6
1/6
1/6
1/6
1/6
1/6
1/61/6
7 8 9
6 5 4
1 2 3
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Model of Snakes and Ladders(S&L)
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The previous “brute-force” listing of all the favorable rolls might be possible for a small game like that.
But what about computing the probability of reaching square 50 in this ?
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ContentsNumbers and History of Chance Relative Frequency and In Class ExperimentDefinition of ProbabilityExercisesCourse Information
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Course Information
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NOT REQUIRED MATERIAL
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Course InformationWebsite link
ContainsLinks to lecturesGrading policyReading materialTake-home assignmentsProject details
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http://www.isaid.rice.edu/elec281.htm
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Calender
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Total number of weeks = 15
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Evaluation and GradingMini-project 1
Around week 515% towards final grade
Mini-project 2Around week 1015% towards final grade
ProjectChosen around week 6Project reviews begin around week
1260% towards final grade
Other supplemental workReading materialTake home exercises10% towards final grade
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Student SurveyPlease fill out the given questionnaire.This is to help adjust the course to better
suit your requirements.
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