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< First of all, I would like to say Alhamdulillah, for giving me the strength and health to do this project work. Not forgotten my parents for providing everything, such as money, to buy anything that are related to this project work and their advise, which is the most needed for this project. Internet, books, computers and all that. They also supported me and encouraged me to complete this task so that I will not procrastinate in doing it. Then I would like to thank my teacher, Madam Zaiton for guiding me and my friends throughout this project. We had some difficulties in doing this task, but she taught us patiently until we knew what to do. She tried and tried to teach us until we understand what we supposed to do with the project work. Last but not least, my friends who were doing this project with me and sharing our ideas. They were helpful that when we combined and discussed together, we had this task done.

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ô The aims of carrying out this project work are: ëto apply and adapt a variety of problem-solving strategies to solve problems; ëto improve thinking skills; ëto promote effective mathematical communication; ëto develop mathematical knowledge through problem solving In a way that increases students¶ interest and confidence; ëto use the language of mathematics to express mathematical ideas precisely; ëto provide learning environment that stimulates and enhances effective learning; ë to develop positive attitude towards mathematics.

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PART ONE

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Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. Probability has a dual aspect: on the one hand the probability or likelihood of hypotheses given the evidence for them and on the other hand the behavior of stochastic processes such as the throwing of dice or coins. The study of the former is historically older in, for example, the law of evidence, while the mathematical treatment of dice began with the work of Pascal and Fermat in the 1650s. Probability is distinguished from statistics. While statistics deals with data and inferences from it, (stochastic) probability deals with the stochastic (random) processes which lie behind data or outcomes. HISTORY - ORIGINS Ancient and medieval law of evidence developed a grading of degrees of proof, probabilities, presumptions and half-proof to deal with the uncertainties of evidence in court. In Renaissance times, betting was discussed in terms of odds such as "ten to one" and maritime insurance premiums were estimated based on intuitive risks, but there was no theory on how to calculate such odds or premiums. The mathematical methods of probability arose in the correspondence of Pierre de Fermat and Blasé Pascal (1654) on such questions as the fair division of the stake in an interrupted game of chance. Christian Huygens (1657) gave a comprehensive treatment of the subject. IMPORTANCE OF PROBABILITY IN LIFE I will assume that you are referring to probability theory. Statistics is based on an understanding of probability theory. Many professions require basic understanding of statistics. So, in these cases, it is important. Probability theory goes beyond mathematics. It involves logic and reasoning abilities. Marketing and politics have one thing in common, biased statistics. I believe since you are exposed to so many statistics, a basic understanding of this area allows more critical thinking. The book "How to lie with statistics" is a classic and still in print. So, while many people would probably say that probability theory has little importance in their lives, perhaps in some cases if they knew more, it would have more importance.

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0

½ m and and their cognates in other modern languages derive from medieval learned Latin and , deriving from Cicero and generally applied to an opinion to mean or .

ô Ancient and medieval law of evidence developed a grading of degrees of proof, probabilities, presumptions and half-proof to deal with the uncertainties of evidence in court. In Renaissance times, betting was discussed in terms of odds such as "ten to one" and maritime insurance premiums were estimated based on intuitive risks, but there was no theory on how to calculate such odds or premiums. The mathematical methods of probability arose in the correspondence of Pierre de Fermat and Blasé Pascal (1654) on such questions as the fair division of the stake in an interrupted game of chance. Christiaan Huygens (1657) gave a comprehensive treatment of the subject.

Jacob Bernoulli's < (posthumous, 1713) and Abraham de Moivre's (1718) put probability on a sound mathematical footing, showing how to calculate a wide range of complex probabilities. Bernoulli proved a version of the fundamental law of large numbers, which states that in a large number of trials, the average of the outcomes is likely to be very close to the expected value - for example, in 1000 throws of a fair coin, it is likely that there are close to 500 heads (and the larger the number of throws, the closer to half-and-half the proportion is likely to be).

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The power of probabilistic methods in dealing with uncertainty was shown by Gauss's determination of the orbit of Ceres from a few observations. The theory of errors used the method of least squares to correct error-prone observations, especially in astronomy, based on the assumption of a normal distribution of errors to determine the most likely true value. Towards the end of the nineteenth century, a major success of explanation in terms of probabilities was the Statistical mechanics of Ludwig Boltzmann and J. Willard Gibbs which explained properties of gases such as temperature in terms of the random motions of large numbers of particles. The field of the history of probability itself was established by Isaac Todhunter's monumental m m (1865).

Ú Probability and statistics became closely connected through the work on hypothesis testing of R. A. Fisher and Jerzy Neyman, which is now widely applied in biological and psychological experiments and in clinical trials of drugs. A hypothesis, for example that a drug is usually effective, gives rise to a probability distribution that would be observed if the hypothesis is true. If observations approximately agree with the hypothesis, it is confirmed, if not, the hypothesis is rejected. The theory of stochastic processes broadened into such areas as Markov processes and Brownian motion, the random movement of tiny particles suspended in a fluid. That provided a model for the study of random fluctuations in stock markets, leading to the use of sophisticated probability models in mathematical finance, including such successes as the widely-used Black-Scholes formula for the valuation of options. The twentieth century also saw long-running disputes on the interpretations of probability. In the mid-century frequentism was dominant, holding that probability means long-run relative frequency in a large number of trials. At the end of the century there was some revival of the Bayesian view, according to which the fundamental notion of probability is how well a proposition is supported by the evidence for it. The mathematical treatment of probabilities, especially when there are infinitely many possible outcomes, was facilitated by Kolmogorov's axioms (1931).

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< First of all, I would like to say Alhamdulillah, for giving me the strength and health to do this project work. Not forgotten my parents for providing everything, such as money, to buy anything that are related to this project work and their advise, which is the most needed for this project. Internet, books, computers and all that. They also supported me and encouraged me to complete this task so that I will not procrastinate in doing it. Then I would like to thank my teacher, Madam Zaiton for guiding me and my friends throughout this project. We had some difficulties in doing this task, but she taught us patiently until we knew what to do. She tried and tried to teach us until we understand what we supposed to do with the project work. Last but not least, my friends who were doing this project with me and sharing our ideas. They were helpful that when we combined and discussed together, we had this task done.

m

ô The aims of carrying out this project work are: ëto apply and adapt a variety of problem-solving strategies to solve problems; ëto improve thinking skills; ëto promote effective mathematical communication; ëto develop mathematical knowledge through problem solving In a way that increases students¶ interest and confidence; ëto use the language of mathematics to express mathematical ideas precisely; ëto provide learning environment that stimulates and enhances effective learning; ë to develop positive attitude towards mathematics.

Ú

m

PART ONE

%

m

3

¬

Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. Probability has a dual aspect: on the one hand the probability or likelihood of hypotheses given the evidence for them and on the other hand the behavior of stochastic processes such as the throwing of dice or coins. The study of the former is historically older in, for example, the law of evidence, while the mathematical treatment of dice began with the work of Pascal and Fermat in the 1650s. Probability is distinguished from statistics. While statistics deals with data and inferences from it, (stochastic) probability deals with the stochastic (random) processes which lie behind data or outcomes. HISTORY - ORIGINS Ancient and medieval law of evidence developed a grading of degrees of proof, probabilities, presumptions and half-proof to deal with the uncertainties of evidence in court. In Renaissance times, betting was discussed in terms of odds such as "ten to one" and maritime insurance premiums were estimated based on intuitive risks, but there was no theory on how to calculate such odds or premiums. The mathematical methods of probability arose in the correspondence of Pierre de Fermat and Blasé Pascal (1654) on such questions as the fair division of the stake in an interrupted game of chance. Christian Huygens (1657) gave a comprehensive treatment of the subject. IMPORTANCE OF PROBABILITY IN LIFE I will assume that you are referring to probability theory. Statistics is based on an understanding of probability theory. Many professions require basic understanding of statistics. So, in these cases, it is important. Probability theory goes beyond mathematics. It involves logic and reasoning abilities. Marketing and politics have one thing in common, biased statistics. I believe since you are exposed to so many statistics, a basic understanding of this area allows more critical thinking. The book "How to lie with statistics" is a classic and still in print. So, while many people would probably say that probability theory has little importance in their lives, perhaps in some cases if they knew more, it would have more importance.

m

0

½ m and and their cognates in other modern languages derive from medieval learned Latin and , deriving from Cicero and generally applied to an opinion to mean or .

ô Ancient and medieval law of evidence developed a grading of degrees of proof, probabilities, presumptions and half-proof to deal with the uncertainties of evidence in court. In Renaissance times, betting was discussed in terms of odds such as "ten to one" and maritime insurance premiums were estimated based on intuitive risks, but there was no theory on how to calculate such odds or premiums. The mathematical methods of probability arose in the correspondence of Pierre de Fermat and Blasé Pascal (1654) on such questions as the fair division of the stake in an interrupted game of chance. Christiaan Huygens (1657) gave a comprehensive treatment of the subject.

Jacob Bernoulli's < (posthumous, 1713) and Abraham de Moivre's (1718) put probability on a sound mathematical footing, showing how to calculate a wide range of complex probabilities. Bernoulli proved a version of the fundamental law of large numbers, which states that in a large number of trials, the average of the outcomes is likely to be very close to the expected value - for example, in 1000 throws of a fair coin, it is likely that there are close to 500 heads (and the larger the number of throws, the closer to half-and-half the proportion is likely to be).

m

The power of probabilistic methods in dealing with uncertainty was shown by Gauss's determination of the orbit of Ceres from a few observations. The theory of errors used the method of least squares to correct error-prone observations, especially in astronomy, based on the assumption of a normal distribution of errors to determine the most likely true value. Towards the end of the nineteenth century, a major success of explanation in terms of probabilities was the Statistical mechanics of Ludwig Boltzmann and J. Willard Gibbs which explained properties of gases such as temperature in terms of the random motions of large numbers of particles. The field of the history of probability itself was established by Isaac Todhunter's monumental m m (1865).

Ú Probability and statistics became closely connected through the work on hypothesis testing of R. A. Fisher and Jerzy Neyman, which is now widely applied in biological and psychological experiments and in clinical trials of drugs. A hypothesis, for example that a drug is usually effective, gives rise to a probability distribution that would be observed if the hypothesis is true. If observations approximately agree with the hypothesis, it is confirmed, if not, the hypothesis is rejected. The theory of stochastic processes broadened into such areas as Markov processes and Brownian motion, the random movement of tiny particles suspended in a fluid. That provided a model for the study of random fluctuations in stock markets, leading to the use of sophisticated probability models in mathematical finance, including such successes as the widely-used Black-Scholes formula for the valuation of options. The twentieth century also saw long-running disputes on the interpretations of probability. In the mid-century frequentism was dominant, holding that probability means long-run relative frequency in a large number of trials. At the end of the century there was some revival of the Bayesian view, according to which the fundamental notion of probability is how well a proposition is supported by the evidence for it. The mathematical treatment of probabilities, especially when there are infinitely many possible outcomes, was facilitated by Kolmogorov's axioms (1931).

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D

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