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1 PROBABILITY THEORY FOR EE 150 Background for coin toss problems and chemical reaction simulations Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to description of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. In probability theory, an event is a set of outcomes (a subset of the of a larger set which is usually defined as the set of possible outcomes of interest ) to which a probability is assigned. Typically, when the set of possible outcomes is finite, any subset of the set of possible outcomes is an event ( i . e . all elements of the of the set of possible outcomes are defined as events). Probability plays a role in many sectors of modern life. Most of us play games of chance in some way or other. This may vary from a simple board game to playing games in a casino for money. In corporate life, actuaries are employed to assess risk of almost every kind so that insurance companies make profits on the premiums they set. There are a number of ways to try to determine the probability of an event. Here we use probability trees. This method is useful when we have basic activities with known theoretical probabilities, such as rolling a dice, tossing a coin, drawing a card from a deck or taking a ball from a barrel. When these activities are repeated we can find the probability of an event such as getting three sixes in three rolls of a dice, by constructing a probability tree. Probability trees can also be used in situations where two different basic events are combined and to compute conditional probability. But first we will just be concerned with learning how to construct and use probability trees when one activity is repeated. Along the way one should always look for patterns in your work. Here we will look for generalizations. What will happen when a coin is tossed n times? What is the likelihood of getting precisely one Tail? Two? These generalizations are basically like finding algebraic patterns but involve slightly more sophistication. Part of the reason for looking for these generalizations is that this is an important part of problem
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This note was uploaded on 04/12/2009 for the course EE 150 taught by Professor Dr.burke during the Fall '08 term at USC.

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