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Best Q2 from Project 1, Nov 2011 Best Q21 from project 1, Nov 2011 Introduction: For the site which demonstrates the imaginative use of random numbers in such a way that it could be used explain the details of the use of random numbers to an entire class, we chose a site that is based on the Infinite Monkey Theorem. This theorem hypothesizes that if you put an infinite number of monkeys at typewriters/computer keyboards, eventually one will type out the script of a Shakespearean work. This theorem asserts nothing about the intelligence of the one random monkey that eventually comes up with the script. It may be referred to semi-seriously when justifying a brute force method (the statement to be proved is split up into a finite number of cases and each case has to be checked to see if the proposition in question holds true); the implication is that, with enough resources thrown at it, any technical challenge becomes a “one-banana problem” (the idea that trained monkeys can do low level jobs). The site we chose is: http://www.vivaria.net/experiments/notes/publication/NOTES_EN.pdf. It was created by researchers at Paignton Zoo and the University of Plymouth, in Devon, England. They carried out an experiment for the infinite monkey theorem. The researchers reported that they had left a computer keyboard in the enclosure of six Sulawesi Crested Macaques for a month; it resulted in the monkeys produce nothing but five pages consisting largely of the letter S (shown on the link stated above). The monkeys began by attacking the keyboard with a stone, and followed by urinating on the keyboard and defecating on it. Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. Lastly, in reality, monkeys at a typewriter are a very poor metaphor for random data. A monkey could be taught how to type simple words, or be influenced by things in the environment, or use their own intellect to produce non-random results. In the infinite monkey theorem, it is important to know that the source of the data has to be truly random. That is, all letters have the same probability of being typed regardless of what letters come before them, thus the monkeys are only used to demonstrate a source of random data. Analysis: 1. Proof of Theorem: Firstly, if two events are statistically independent, then the probability of both events happening equals the product of the probabilities of each one happening independently i.e. Pr (1) x Pr (2) The average keyboard has roughly 50 keys (including non-letter keys). If one would assume that the keys are pressed randomly and independently, then the chance of the first letter 'm' typed of a word, say 'monkey', is 1/50 and the chance of the second letter typed is 'o' is also 1/50, and so on for the rest. Therefore the probability of the first n letters 1 MS1: Ciarán Porter, Gabriele Cecchi; Darragh Austin. matching of the word is: (1/50) x (1/50) x (1/50)...= (1⁄50)(퐧) Thus, from the above probability, the probability of not typing 'monkey' with 6 letters in a block of n letters is 1 -(1⁄50)(6). Furthermore, as each block is typed independently of the others, the probability of not typing 'monkey' in any of the first n blocks which contain the same amount of letters is: 푿풏 = (1 − (1⁄506))풏 As n approaches infinity, Xn approaches zero. However, in real life the results are reversed for the physical numbers of monkeys typing for physical lengths of time. This is because if one ignores capitalization, punctuation, and spacing, the probability of a monkey typing the first letter of a Shakespearean work correctly is 1/26. The probability of typing the first n letters is: (1/26)n. Thus, because the probability shrinks exponentially, the probability of a monkey typing a work of Shakespeare is barely conceivable it is that small. Reasons for picking this site: • One reason for choosing this site was because as we found out by research, the infinite monkey theorem is considered to be part of modern popular culture. We thought this was both interesting and useful as a way of demonstrating random numbers to a class, due to its popularity. We found that the theorem was mentioned in literature (“The Hitchhiker’s Guide to the Galaxy”), and in sitcoms such as “The Simpsons”, “Family Guy”, and “That 70’s Show”. As a result of the theorem being referenced in such well known TV series and books, a class would be able to recall where the theorem was mentioned and would be more capable of understanding how random numbers work using the references in these texts as related examples, without having to make such examples up themselves. • In addition, though the monkeys in the theorem are only used as metaphors, they still made the random numbers more interesting and understandable than say an abstract method of learning random numbers such as creating large amounts of random numbers on an Excel Spreadsheet without a believable or interesting example to use them on. • Also, we found that the differences between the probabilities of (1⁄50)(n) and (1/26)n was quite astonishing, but understandable. As a result this was one of the reasons we chose this site as it showed clearly, by a huge amount the differences between the two, which we thought a class would also be able to understand clearly and consequently comprehend the differences between both of the probabilities. • Lastly, we chose this site as it described an example which we found to be simple to understand and fascinating to read. This was due to the simple mathematics involved in calculating the probabilities and also the way the example was realistic in the way that a monkey could quite possibly type out a work by Shakespeare. Thus, as students, since we believed it helped our learning of random numbers, it would be a good example to demonstrate the imaginative use of random numbers to a class. Conclusion: To conclude, from researching this site and its example of random numbers, it gave us an understanding and knowledge of random numbers that allowed us to competently complete question 1 of the assignment. It also aided our learning on the application of random numbers and its usage in society. We hope that this investigation will prove beneficial towards improving our general understanding of random number/probability theory. .
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