1. Independence, Dependence and History of Numbers 1 Krishna.V.Palem Kenneth and Audrey Kennedy Professor of Computing Department of Computer Science, Rice University

2. Content Intuition to Monty Hall problem Notion of distribution In class exercise Review of random variables Expectation 2

3. Exercise 8 The Monty Hall problem is a probability puzzle based on the American television game show Let's Make a Deal. Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say Number 3, which has a goat. He then says to you, "Do you want to pick door Number 2?" Is it to your advantage to switch your choice? 3

4. Developing the intuition Do you think the host opening the door is independent of your choice of the door? Ans: NO! Hint: If you choose a door with a goat, the host MUST open the other door with the goat If you choose a door with a car, the host can pick one of the 2 doors with the goat. So the host opening the door is DEPENDENT on your choice of the door!! Now try to solve the problem!! 4

5. Intuition Continued Imagine you doing this experiment 99 times. Suppose you are asked to choose a door. No. of times you will choose a door with a goat = (99*2)/3 = 66 times. No. of times you will choose a door with a car = (99*1)/3 = 33 times. Case 1: You don’t switch No. of times you win the car after host opens door with a goat = No. of times you chose a door with a car in the first place = 33 times. Probability of winning without switching = 33/99 = 1/3. 5

6. Intuition Continued Case 2: You switch Important Intuition Case 2.1: If you choose a door with a goat, The host MUST choose the other door with the goat As a result, the third unopened door MUST have the car. Clearly, switching = Winning the car Case 2.2: If you choose a door with a car, Clearly, switching does not help at all! Therefore, the number of times you will win if you switch = Number of times you choose a door with a goat in the first place = 66 times!!! (2x33 times) Result: Switching doubles your chance of winning!! 6

7. Intuition Continued Case 2: You switch Important Intuition Case 2.1: If you choose a door with a goat, The host MUST choose the other door with the goat As a result, the third unopened door MUST have the car. Clearly, switching = Winning the car Case 2.2: If you choose a door with a car, Clearly, switching does not help at all! Therefore, the number of times you will win if you switch = Number of times you choose a door with a goat in the first place = 66 times!!! (2x33 times) Result: Switching doubles your chance of winning!! 7

8. Content Intuition to Monty Hall problem Notion of distribution In class exercise Expectation 8

9. The notion of a distribution A distribution is a function defined on the random variable that gives the value of the probability of the random variable taking a particular value Xp(X) 11/6 2 3 4 5 6 Why do we need a distribution? Compactness It clearly shows that time and again, number representations have evolved into more and more compact representations from tally marks to decimal representation. Similarly, a distribution is a compact way to represent a random variable and all the possible outcomes. 9

10. Uniform distribution Lets look at the following game of dice problem. For a fair die, the probability seeing a 1, 2, all the way to 6 is 1/6. 10 Xp(X) 11/6 2 3 4 5 6 An example of a compact representation of this would be an equation P(X=i) = 1/6 for all i={1,2,3,4,5,6} is a distribution. Such a distribution is called “Uniform distribution” If there is a fair die which has N faces, then P(X=i) = 1/N for all i={1,2,3,….,N}

11. Geometric distribution Suppose you throw a die till you see number 1 after which you stop. Here we want to find the probability of throwing the die N times 11 Xp(X) 11/6 All other number 5/6 P(just one throw) = P(getting a 1 in first throw) = 1/6 P(just two throws) = P(getting number other than 1 in first throw)* P(getting number 1 on second) = (5/6)*(1/6) P(just N throws) = P(getting number other than 1 in first N-1 throws)*P(getting 1 on Nth throw) = (5/6) N-1 (1/6) This is a “geometric distribution”

12. Distribution and Compactness So which one is a convenient representation? 12 No of Rolls P(X) 11/6 25/36 325/216 4125/1296 5625/7776 …… …… OR P(X) = (5/6) X-1 (1/6) History of numbers: Numbers have evolved into more and more succinct representations. XXXVI10.34

13. Objects and Fingers 13 Let us consider this hunter spots a herd of deer The hunter has to convey this information to his group. He uses two of his fingers to signal this Thus the hunter has been able to express a count of the animals. This abstract form of representation of information about surroundings is the Number. Each finger is an abstracted representation of a deer.

14. Objects and Fingers 14 A forager has collected a certain number of fruit The forager showsto denote the five fruit Note that both theandare represented by fingers a “common” representation This common representation of magnitude of any quantity is the “Number” Let us consider another example Used to count two entirely different objects

15. Number – An Abstract concept 15 A number can represent anything Number of animals Height of a mountain Number of trees in a forest Distance between two places The literal form or symbol in which a number is represented is called a “numeral” In our previous case In this lecture we will learn how the number and its representation evolved in time and served as a basis for representing chance

16. The concept of Number 16 http://www.trans4mind.com/personal_development/astrology/numbers.htm Bird sees a ‘group’ of hunters nearing its nest One of the hunter hides behind the bush, while the rest leave. Bird sees the ‘group’ of hunters leaving. Assumes it is safe to return to its nest, thus being in a vulnerable position. Number is one of the most momentous idea in the history of human development. One mystery is the origin of numbers and how they come about. For example, it is said of birds, that they can count up to three, but not beyond. More than three is a “group”.

17. Origin of Numbers The primitive notion of numbers appears to have evolved from the many physical contrasts prevalent in nature 17

18. Fingers and Tally marks 18 In the previous example we saw that the hunter could specify the 2 deer with Consider this case: It was not possible to represent an arbitrary quantity using fingers. Therefore, humans started representing objects using straight lines instead of fingers. The big difference being they could draw any number of straight lines. Instead of drawing straight lines adjacent to each other. For ease of counting the lines were grouped together. Thus emerged the concept of “The Tally Marks” It was necessary to use some other form of representation. First Number representation – “Tally Marks”

19. The Tally Marks One of the most basic form of representation of quantities. It consists of only one symbol = A Numeral is the actual representation of a number using these symbols. This system is still used widely where frequent updates of a certain quantity is necessary. A certain rather disturbing example is shown in the picture. 20000 BC 3000BC500BC300BC500AD1000AD 19 234 5 STOPWATCH

20. Content 20 1. Numbers and Chance 2. What is a number? Why was it invented ? 3. The first forms of numbers. 4. From numbers emerges the concept of Arithmetic – the notion of manipulation of numbers. 5. Numbers, Arithmetic and Value to Society 6. Evolution of number representation. 7. The Story of Zero 8. Effect of numbers 9. Fractional numbers 10. Conclusion

21. Number Systems from different parts of the world At around 3000BC, there were three major civilizations across the world that were flourishing. Babylonian Egyptian Tally Marks could not satisfy the needs of humankind for long. Each of these civilizations developed their own number system. The numerals were different and also the value system that was adopted was different. 20000BC 3000 BC 500BC300BC500AD1000AD 21

22. The Babylonian Number system The Babylonians lived in Mesopotamia, a fertile plain between the Tigris and Euphrates rivers. 3400 BC: The Egyptians and Babylonians were first recorded as using the natural numbers and rational numbers. The base of a number system is the number of symbols available for representation. The modern day numbers are a base-10 system, because we have 10 different symbols for all our numbers. Babylonians had a base of 60. That means they had 60 symbols. 20000BC 3000 BC 500BC300BC500AD1000AD 22 0123456789 …. Notice the white space for a zero !! 01 2 345859

23. Sumerian/ Babylonian Numerals 20000BC 3000 BC 500BC300BC500AD1000AD 23 Though we mentioned that the Babylonians had a base of 60 which means that they have to remember 60 different symbols to use numbers, they had invented a clever way of creating all the 60 numerals from just two symbols. And They needed 59 numerals to represent all the numbers from 1 and 59. So they had a symbol for 10 and another one for 1. Thus the number of times each one of them is repeated gives the numeral for that particular number. Now let us create the numeral 7 Thus to create a numeral for 7, we need to combine 7 of the symbols for 1.

24. Sumerian/ Babylonian Numerals 24 20000BC 3000 BC 500BC300BC500AD1000AD Thus to create a symbol for 23, we need two symbols of ‘10’ and three of ‘1’. Now let us consider another example Let us create the numeral 23 Exercise: Create the numeral for the number 33 using the Babylonian symbols Solution:

25. Sumerian/ Babylonian Numerals 25 To create other numerals there are some simple rules that have to be followed. 1.When we are stacking symbols for 1 to create numerals, each stack should have at most three in each row. For example,To create the number 3 We stack all the three symbols in a single row. To create the number 5 We stack three in the first row and then the remaining two in the next row. Similarly when we stack the symbols for 10, the symbols are stacked in the same way except diagonally. Can you infer from the following ?

26. Constructing bigger numerals from small numerals 20000BC 3000 BC 500BC300BC500AD1000AD 26 Thus we obtain the complete set of symbols for all the 59 numbers. 0 Now as we have all the numerals for the number system, we need to understand how to write to write a number.

27. Positional Significance 27 Consider the number 4256. This is a numeral. What is the value of this numeral ? How is it evaluated? 4256 = 4*1000 + 2*100 + 5*10 + 6 Positional significance Positional significance is the mechanism by which a symbol is elevated in it’s value to easily create bigger numbers. This is done by writing the symbols adjacent to each other to create bigger numerals. Powers of 10 Exercise: What would 4256 represent if the positional significance would have been as the powers of 2?

28. Sumerian/ Babylonian Numbers 28 For example, let us try writing in Babylonian system, the value 4256 So, 4256 can be written as 4256 = 1*3600 + 10*60 + 56*1 = 1*60 2 + 10*60 1 + 56*60 0 As the Babylonian system has a base of 60, the positional significance of each symbol varies with as a power of 60. Now, we have to represent this in terms of the Babylonian numerals Solution: Exercise: Represent the quantity 2764 in Babylonian number system.

29. A quick recap 29 TopicTally MarksConventionalSumerian Symbol0,1,2,3…,9 Numerals4256 Base11060 Positional significance Powers of 1Powers of 10Powers of 60

30. Problems in Sumerian/ Babylonian System 30 Now there is a potential problem with the system. Using this number system let us represent the two numbers 61 and 2. First 61 = 1*60 +1*1 is represented as And 2 = 2*1 is represented as The only difference being the space between the symbols. A much more serious problem was the fact that there was no symbol for zero. They have exactly the same representation and now there was no way that spacing could help. Let us see for ourselves. Let us represent the numbers 1 and 60. 1 = 1*160= 1*60

31. Content Intuition to Monty Hall problem Notion of distribution In class exercise Expectation 31

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