notes09c1 - 1. Introduction to Probability 1.1 Definitions...

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1. Introduction to Probability 1.1 Definitions of Probability You are the product of a random universe. From the Big Bang to your own conception and birth, random events have determined who we are as a species, who you are as a person, and much of your experience to date. Ironic therefore that we are not well-tuned to understanding the randomness around us, perhaps because millions of years of evolution have cultivated our ability to see regularity, certainty and deterministic cause-and-effect in the events and environment about us. We are good at finding patterns in numbers and symbols, or relating the eating of certain plants with illness and others with a healthy meal. In many areas, such as mathematics or logic, we assume we know the results of certain processes with certainty (e.g., 2+3=5), though even these are often subject to assumed axioms. Most of the real world, however, from the biological sciences to quantum physics 1 , involves variability and uncertainty. For example, it is uncertain whether it will rain tomorrow; the price of a given stock a week from today is uncertain; the number of claims that a car insurance policy holder will make over a one-year period is uncertain. Uncertainty or “randomness" (i.e. variability of results) is usually due to some mixture of at least two factors including: (1) variability in populations consisting of animate or inanimate objects (e.g., people vary in size, weight, blood type etc.), and (2) variability in processes or phenomena (e.g., the random selection of 6 numbers from 49 in a lottery draw can lead to a very large number of different outcomes). Which of these would you use to describe the fluctuations in stock prices or currency exchange rates? Variability and uncertainty in a system make it more difficult to plan or to make decisions without suitable tools. We cannot eliminate uncertainty but it is usually possible to describe, quantify and deal with variability and uncertainty using the theory of probability. This course develops both the mathematical theory and some of the applications of probability. Applications are far-reaching, from
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This note was uploaded on 02/11/2010 for the course ART AFM101 taught by Professor Mr.lushman during the Spring '10 term at University of Toronto- Toronto.

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notes09c1 - 1. Introduction to Probability 1.1 Definitions...

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