Chance and Probability 1 Krishna.V.Palem Kenneth and Audrey Kennedy Professor of Computing...

Chance and Probability

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Krishna.V.PalemKenneth and Audrey Kennedy Professor of ComputingDepartment of Computer Science, Rice University

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.

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

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”

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.

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

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

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}

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

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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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

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

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

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.

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}

Probability of more than one event

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RecallConsider an event space

Event 1

Event 2

Event 3

Event 4

Favorable events


No. of favorableevents

Total no. of events

Event Probabilities

Event 1 p1

Event 2 p2

Event 3 p3

Event 4 p4

Given this


21 pp

This is the probability of event 1 OR event 2


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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

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

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

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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

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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

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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Course Information

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NOT REQUIRED MATERIAL

Course InformationWebsite link

ContainsLinks to lecturesGrading policyReading materialTake-home assignmentsProject details

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http://www.isaid.rice.edu/elec281.htm

Calender

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Total number of weeks = 15

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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Chance and Probability 1 Krishna.V.Palem Kenneth and Audrey Kennedy Professor of Computing...

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