Chance and Probability 1 Krishna.V.Palem Kenneth and Audrey Kennedy Professor of Computing Department of Computer Science, Rice University

Contents Numbers and History of Chance Relative Frequency and In Class Experiment Events and Definition of Probability Exercises Course Information 2

Contents Numbers and History of Chance Relative Frequency and In Class Experiment Events and Definition of Probability Exercises Course Information 3

Cause and Effect of numbers 4 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 5 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 complete one cycle. Numbers could now quantify the precise amount of time taken for 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 6 The entire gambling industry is based on numbers. Slot machines are designed in such a way that the casino always is at an advantage. This is done by calculating the average chance of winning that does not cause the casino to 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 7 The advent of ‘numbers’ provided people an opportunity to quantify chance. However, not until the 17th century did these concerns lead to attempts to understand the principles mathematically using numbers.

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 8 33 Million BC 250,000BC3500BC3000BC1100BC400BC

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 9 250,000 BC 33 Million BC3500BC3000AD1100BC400BC The hunter-gatherer lifestyle of early cultures was predicated on risk and reward. like mining and fishing today

Chance in Ancient World As 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 10

Explaining Chance What 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. 11

The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern - Keith Devlin 12

The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern - Keith Devlin 13

Contents Numbers and History of Chance Relative Frequency and In Class Experiment Definition of Probability Exercises Course Information 14

Events and Outcomes Consider an experiment where one dice is rolled Can show 6 different values on the top side Top side Hence this experiment is said to have 6 different possible events

Events and Outcomes (contd.) Consider the same experiment where one dice is rolled Top side 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 of the 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)? 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 18 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 rollSquare landedFavorable Outcome (6 ) Relative frequency = Number of favorable outcomes Total number of outcomes

Contents Numbers and History of Chance Relative Frequency and In Class Experiment Definition of Probability Exercises Course Information 19

Probability 20 The classical definition of probability Pierre Simon Laplace The 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 reason believe that one event is more likely to occur than another If there is no reason believe that one event is more likely to occur than another Probability of favorable events Favorable events No. of favorable events Total no. of events

Fairness In 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 ?

Generalizing Probability 22 Consider another event space Event 1 Event 2 Event 3 Event 4 Let us assume that each event is differently likely to occur Let us represent as a list of magnitude of “likeliness” EventLikeliness Event 1p1p1 Event 2p2p2 Event 3p3p3 Event 4p4p4

Generalizing Probability (Contd.) 23 EventLikeliness Event 1p1p1 Event 2p2p2 Event 3p3p3 Event 4p4p4 Let us look at these values. If these values have the following properties 1.All p i are in the range [0,1] 2.The sum of all p i = 1. then these values are called the probabilities of these events. For example, Event Probabilities Event Event Event Event Satisfies both the conditions This means that event 1 occurs 1 out of 4 times … etc.

Specification 24 Consider the event space Event 1 Event 2 Event 3 Event 4 Event Probabilities Event 1p1p1 Event 2p2p2 Event 3p3p3 Event 4p4p4 Complete specification of the behavior of the experiment Both these together

25 Now we have this relationship Event Event 1 Event 2 Event 3 Event 4 Probabilities p1p1 p2p2 p3p3 p4p4 This can be defined concisely as a function whose independent variable represents the event The dependent variable is the value of the probability Let the variable ‘x’ represent the event xEvent 1Event 1 2Event 2 3Event 3 4Event 4 Probability (Event i) Here ‘x’ is called a random variable. Where i={1,2,3,4}

Probability of more than one event 26 Recall Consider an event space Event 1 Event 2 Event 3 Event 4 Favorable events Probability of favorable events No. of favorable events Total no. of events Event Probabilities Event 1p1p1 Event 2p2p2 Event 3p3p3 Event 4p4p4 Given this Probability of favorable events This is the probability of event 1 OR event 2

Contents Numbers and History of Chance Relative Frequency and In Class Experiment Definition of Probability Exercises Course Information 27

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)?

Solution : Hint 29 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 rollSquare landedFavorable Outcome (6 or 8) Relative frequency = Number of favorable outcomes Total number of outcomes

Building a model An 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 /6

31 Solution: Probability of reaching 6 via 5 = 1/6 Probability of reaching 6 via 4 = 1/6 Total probability of reaching 6 from 4 OR 5 = 1/6 + 1/6 = 1/ /6

32 This is the complete model of the scaled down snakes and ladder game /

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Model of Snakes and Ladders(S&L) 34 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 ?

Contents Numbers and History of Chance Relative Frequency and In Class Experiment Definition of Probability Exercises Course Information 35

Course Information 36 NOT REQUIRED MATERIAL

Course Information Website link Contains Links to lectures Grading policy Reading material Take-home assignments Project details 37

Calender 38 Total number of weeks = 15

Evaluation and Grading Mini-project 1 Around week 5 15% towards final grade Mini-project 2 Around week 10 15% towards final grade Project Chosen around week 6 Project reviews begin around week 12 60% towards final grade Other supplemental work Reading material Take home exercises 10% towards final grade 39

Student Survey Please fill out the given questionnaire. This is to help adjust the course to better suit your requirements. 40