Feedback on Project and Mini-Project I

& Statistics

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

ContentsFeedback on ProjectsDiscussion of Mini-Project I solutionHistory of StatisticsBasic terms involved in StatisticsSampling

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Biography of Blaise PascalBlaise Pascal was a French

mathematician, physicist, inventor, writer and Catholic philosopher.

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Pascal's earliest work was in the natural and applied sciences Made important contributions to the study of fluids, clarified the concepts of pressure and vacuum

Pascal was a mathematician of the first order. helped create two major new areas of research.

wrote a significant treatise on the subject of projective geometry at the age of sixteen

corresponded with Pierre de Fermat on probability theory,

strongly influenced the development of modern economics and social science.

Biography of Pierre de FermatPierre de Fermat was a French lawyer at the Parlement

of Toulouse, France, and a mathematician He is given credit for early developments that led to

infinitesimal calculus. In particular, he is recognized for his discovery of an

original method of finding the greatest and the smallest ordinates of curved lineswhich is analogous to that of the then unknown differential

calculus, He made notable contributions to analytic geometry,

probability, and optics as well as his research into number theory.

He is best known for Fermat's Last Theorem, It was described in a note at the margin of a copy of

Diophantus' Arithmetica.

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Mini-Project DiscussionThe problem being modeled in the mini-project

is the problem of pointsAlso called the problem of division of the stakes

One of the famous problems that motivated the beginnings of modern probability theory in the 17th centuryThrough this discussion Pascal and Fermat

developed concepts that continue to be fundamental in probability to this day.Along with coming up with a convincing, self-consistent

solution to the division of the stakesIt also led Pascal to the first explicit reasoning

about what today is known as an expectation value.

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Mini-Project DiscussionThe problem concerns a game of chance with two

players who have equal chances of winning each round. The players contribute equally to a prize pot, and agree in

advance that the first player to have won a certain number of rounds will collect the entire prize.

Now suppose that the game is interrupted by external circumstances before either player has achieved victory. How does one then divide the pot fairly? \ Inspiration for the people predicting outcomes of the Cricket game

It is tacitly understood that the division should depend somehow on the number of rounds won by each player, such that a player who is close to winning will get a larger part of the pot. But the problem is not merely one of calculation; it also includes

deciding what a "fair" division should mean in the first place.

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Basic Premise of the Problem

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To understand the basic premise of the problem, let us construct an example (albeit imaginary) based on the problem description

Pascal and Fermat are sitting in a cafe in Paris and decide, to play by flipping a coin. If the coin comes up heads, Fermat gets a

point. If it comes up tails, Pascal gets a point. The first to get 10 points wins. They each bet 50 Francs and 'winner takes

all'.

Basic Premise of the Problem

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But then an unexpected thing happens. Fermat is winning, 8 points to 7, when he receives an urgent message that a friend is sick, and he must rush to his home. The carriage man who has delivered the

message offers to take him, but only if they leave immediately.

Of course Pascal understands, but later, in correspondence, the problem arises:

How should the 100 Francs be divided?

Solution to the problem by Fermat

Seeing as I(Fermat) needed only two points to win the game, and you needed 3, I think we can establish that after four more tosses of the coin, the game would have been over

All the possible ways of this happening are as follows:I think you will agree that all of these outcomes are equally likely.

Based on these possible outcomes, I believe that we should divide the stakes by the ration 11:5 in my favorI should receive (11/16)*100 = 68.75 Francs, while you should

receive 31.25 Francs.

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A More General Solution by PascalPascal was able to use it as a starting point for a

developing even slicker computation methods. He reasoned that any outcome that featured two or

more heads turning up meant a win for you (Fermat). The total number of such outcomes is equivalent to

the number of ways to choose two objects from four, plus the number of ways to choose three objects from four,

plus the number of ways to choose four objects from four.

Here the 'events' of the coin coming up in your favor become 'objects' in our counting terminology.

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A More General Solution by PascalLet us denote the number of ways to

choose to choose r objects from n objects as nCr. Thus I think the likelihood that you would have won the game is given by:

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4C2 + 4C3 + 4C4 -----------------------total outcomes

Exercise: Verify that this value is similar to the one derived by Fermat on Slide 17

A More General Solution by Pascal

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Generalizing the problem : Let us suppose that two players are playing, and the game is stopped after a certain number of roundsThe first player needs n more points to win,

while the second needs m more points.We solve this problem by constructing a

Pascal’s triangle as described below The rows of Pascal's triangle are conventionally

enumerated starting with row n = 0 at the top. The entries in each row are numbered from the left

beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows.

Pascal’s TriangleA simple construction of the triangle proceeds in the

following manner. On row 0, write only the number 1. Then, to construct the elements of following rows,

add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present,

substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1,

whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

Expressing the numbers in terms of combinations, we get

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Pascal’s Triangle

A More General Solution by PascalRecall that the first player needs n points to

win, while the second needs m.To compute how the stakes should be divided,

one computes row (n+m) of the triangle, and then adds up the first m entries.

This number is corresponds to the first player's (who needs n points to win) share of the stake.

The remaining entries should be added up to determine the second players share.

14Note: Full solution will be posted online

Exercise: Apply this general proof to solve the specific case (that Fermat requires 2 points to win and Pascal requires 3 points to win) and verify that you

get the same solution as before

ContentsFeedback on ProjectsDiscussion of Mini-Project I solutionHistory of StatisticsBasic terms involved in StatisticsSampling

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Feedback on ProjectsFour main aspects of these projects are:

1. Description of the Problem or the Model2. Problem Statement

Given something Find “best” of something else

3. Evaluation Criteria Prove that your strategy is best, pretty good etc.

Quantifying how good a strategy is

4. Lessons learned

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ModelThere are 3 main characteristics of a model:

1. Constraints A concise list of constraints involved

Constraints which can be subsumed by others should be appropriately identified

Some of the constraints can be slacked/tightened if it can significantly ease the solving of the problem

For example, o in the game of ‘Settlers of Katan’, one of the important

constraint would be that – a player cannot build a new city unless there is a connecting path to one of his existing cities

o in the “Cricket project”, one of the constraint would be that – there can be no more than 300 balls that can be bowled in an innings or there can be no more than 10 wickets that can be taken

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Model2. Configurations or legal states

Presents the values of nodes/variables at a given state

A concise list of nodes/variable that need to be included in a configuration must be identified

For example,o in the game of ‘Settlers of Katan’, the nodes

that can be included in a configuration are initial map position, number of cards of each type etc.

o in the game of ‘Cricket’, the nodes of a configuration would include number of bowls left, number of batsmen left etc.

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Model3. Possibilities for transitions

Defines legal moves between different configurations

For example, in the game of ‘Settlers of Katan’, possibilities of

transitions might include one of the ways to trade-in cardso the actual transition would depend on the roll of

the dice

As a general rule, any model shouldn’t be too cumbersome and should be easy to apply

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Problem StatementProblem statement is a concise description of the issues

that need to be addressed in the attempt to solve the problem.

The primary purpose of a problem statement is to focus the attention of the problem solving team. However, if the focus of the problem is too narrow or the scope of the solution

too limited the creativity and innovation of the solution can be limited.

A good problem statement should have the following aspects:Concise description of the goal

can use the configuration to define goalsDescription of problem to be solved

finding the “best” or “good enough” solution to the problem

This problem statement could be finding a “strategy” for the board games or a good “predictor” for the cricket game

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Evaluation CriteriaEvaluation criteria has two main aspects to

it:Defining the metrics for quantifying your

successUse the metrics to prove that your strategy is the

“best”, “good enough” etc. quantitativelyFramework for evaluation

Can either be experimental or mathematicalFor example, experimental framework might

involve a programming environment while a mathematical framework might involve a proof or a closed form analytical solution

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Lessons LearnedA detailed description of important lessons

learnt in this projectInvolves ways to generalize the solution to

problems broadly in its categoryFor example, the strategy being developed in board

games can be generalized to war strategy simulations

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ContentsFeedback on ProjectsDiscussion of Mini-Project I solutionHistory of StatisticsBasic terms involved in StatisticsSampling

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Birth of StatisticsStatistics arose from the need of states to collect data

on their people and economiesfor administrative purposesstarted in 18th century

Bayes theorem provided the mathematical basis for this branchinitial intuition was given by Francis BaconThomas Bayes provided the first mathematical basis to this

branch of logic

Its meaning broadened in the early 19th century to include the collection and analysis of data in general. today statistics is widely employed in government, business,

and the natural and social sciences.

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Probability Theory Vs Statistics

Probability theory computes the probability that future (and hence presently unknown) samples out of a known population turn out to have stated characteristics

Statistics looks at the present and hence known sample taken out of an unknown population, and makes estimates of what the population is likely to be, compares likelihood of various populations and tells how confident you have a right to be about these estimates

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What is Statistics?Statistics is a mathematical science

pertaining to the collection, analysis, interpretation or explanation, and presentation of data.provides tools for prediction and forecasting

based on data. applicable to a wide variety of academic

disciplinesfrom the natural and social sciences to the

humanities, government and business.

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What is Statistics?

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