IntroductiontoProbability - An Introduction to Probability...

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An Introduction to Probability Theory Santosh Appathurai * The true logic of this world is the calculus of probabilities. Pierre–Simon de Laplace 1 An introduction to this Introduction Very broadly, one could say that probability is the study of averages of phenomena. The basic notion here is that of a random experiment – one whose outcome is not known apriori (Mathspeak for beforehand), but some reasonable predictions can be made on the lines of what it (the outcome) would likely be. The classic examples here would include tossing a coin, rolling a die or picking out a card at random from a well-shuFed pack of cards. This was because in its initial days, probability theory was mainly used to analyze games of chance. To be slightly more precise, it all probably started in ±rance in the mid-17th century, when a nobleman who was rather fond of gambling wrote to two of his compatriots, who happened to be rather fond of numbers (for an elaborate story, see Section 5). ±ollowing its hedonistic beginnings, probability underwent much of the rigor, and soon acquired the polish, that being associated with mathematics and mathematicians begets. In the modern day, probability theory has been used in several important ²elds and has led to numerous advancements in science and technology. This list makes for a very impressive reading – with topics ranging from the foundations of modern physics as we know it, to cryptography, weather predictions, stock markets, going all the way to fantasy football player–prices! The key idea here, is to take into account the information available about a certain system, and make a prediction concerning its behavior under some operating conditions, based on historical con²dence or logical arguments. Indeed, having been brought up with a deterministic view of the laws of physics, one may tend to think that such a theory based on inadequate ²rst principles is not correct . However, it is pertinent to note here that the laws of physics themselves are, but a mathematical description of reality. They merely approximate the system very reliably, under certain operating conditions, to yield results that are within an acceptable error limit. As an * . Some rights reserved 1
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example, consider the motion of a projectile launched with a velocity v , at an angle θ from the ground. as any basic physics textbook would tell, we get the following expression for the range of the projectile d (distance traveled before it hits the ground again): d = ± v 2 g ² sin2 θ (1) For v = 6 ft/s, at an angle θ = 45 o and g = 32 ft/s 2 , one would obtain d =1 . 125 ft. It is safe to say that if one were to ±re a number of such projectiles, not all of them would travel this exact distance! They would all fall in some close neighborhood of 1 . 125 ft. This formula (model) neglects the air–resistance, and assumes that there are going to be no uncertainties in the values of v and θ . Thus, it is important to note that all descriptions of systems (models) are to be accepted only with certain qualifying assumptions.
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This note was uploaded on 09/19/2011 for the course CHE 320 taught by Professor Harris during the Spring '10 term at Purdue University-West Lafayette.

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IntroductiontoProbability - An Introduction to Probability...

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