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Unformatted text preview: IEN 310 Lecture Notes Introduction to Engineering Probability Professor J. Sharit University of Miami Spring 2009 Chapter 1: Experiments, Models, and Probabilities 1. Introduction to the Concept of Probability We generally accept as a given that the probability of an event—that is, the probability of something happening—is a number in the interval [0,1]. One way of viewing this number is as a weight. For example, it seems intuitive to conclude that a randomly selected coin from a jar of coins will have a high probability of being a penny if that jar seems to be filled mostly with pennies. Another view of probability is as a “personal degree of belief” or “knowledge we have about something.” Such “subjective probabilities” imply that different individuals will assign different probabilities to the same event. For example, different weather forecasters may assign different probabilities to the event “it will rain today.” The mathematics of probability, to which both of these views must conform, consists of definitions , axioms , and theorems . The definitions establish the logic of probability, whereas the axioms are facts we accept without proof. The theorems are consequences that follow logically from definitions, axioms, and other theorems. As we shall see, there are only three axioms of probability theory. Fundamental to the subject of probability is the premise that the situations of interest are sufficiently complex to the point where it is unreasonable to assume that we could replicate all the aspects of that situation exactly. Thus probability theory was developed to describe phenomena that cannot be predicted with certainty. Repetitions of the same procedure will therefore lead to different results. The basis for this uncertainty in outcome is the concept of variability . Virtually all system processes exhibit variability. For example, if we examine the thickness of oxide coatings on silicon wafers in the fabrication of integrated circuits, they will be found to differ. Of course, if we can predict the extent of these thicknesses, we are in a position to control the process so that the quality of the end product is not undermined. In functional relationships of the kind we are accustomed to seeing in basic physics, the inputs (for example, speed, mass) are assumed to be known, and result in precise outputs (for example, force). However, even a situation as seemingly simple as flipping a coin has sufficient variability to make it virtually impossible to predict with total certainty what the outcome will be. Although each outcome of this experiment may have a degree of unpredictability associated with it, there may be a consistent pattern that is observed when the experiment is repeated or replicated a large number of times. In the case of flipping a balanced coin, the pattern that emerges as the number of coin flips becomes larger and larger is that a “heads” occurs about half the time....
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 Spring '09

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