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05 Elementary Probability

# 05 Elementary Probability - CS-350 Fundamentals of...

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CS-350: Fundamentals of Computing Systems Page 1 of 15 Lecture Notes © Azer Bestavros. All rights reserved. Reproduction or copying (electronic or otherwise) is expressly forbidden except for students enrolled in CS-350. A Basic Introduction to Probability The study of probability is all about the study of chance—the quantification of the chance or the likelihood that something (e.g. an event, an outcome) will happen. Events and their Probabilities Probability is usually measured in fractions or percentages. For example, a weather forecaster might say that there is a one in five chance (or probability) of rain. This probability is expressed as one fifth or 20%. As another example, consider the outcome of rolling a die. The die has six faces, numbered as 1, 2, 3, 4, 5, and 6. Since it is equally likely that the die will end up with any one of these faces up, the likelihood (or probability) that the outcome is any one of these values is precisely 1/6. In both of the above examples (weather forecast and rolling a die), we associated a probability with a specific event which is the outcome of a specific experiment. For example, the experiment at hand may be checking whether it is raining at a particular locale. The outcome of this experiment is either “raining” or “not raining.” Similarly, the experiment may be rolling a die, in which case the outcome of the experiment is a number between 1 and 6. The maximum probability of any outcome is 1 or 100%. A probability of 1 represents certainty, meaning that the outcome will definitely happen. For example, the probability that the sun will set within the next 24-hour period (in Boston) is 1. A probability of 0 represents certainty as well! It means that the outcome associated with that 0 probability will never happen. The probability of obtaining 7 as the outcome of rolling a six-sided die is 0. To be able to evaluate the likelihood of an event, one needs to figure out the number of ways that the event will materialize relative to all possible outcomes of the experiment. Example: A disk cache holds 50 blocks of data. Assuming that the disk contains 10,000 such blocks, what is the probability that a request to the disk will hit in the disk cache? Solve this problem under the assumption that all disk blocks are equally likely to be accessed. A Motivating Example In a class of N students, what is the probability that two students will have the same birthday? The above question is one of the “classical” questions used to motivate the concepts of event probabilities. To answer this question, we must first make some assumptions so that we are able to

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CS-350: Fundamentals of Computing Systems Page 2 of 15 Lecture Notes © Azer Bestavros. All rights reserved. Reproduction or copying (electronic or otherwise) is expressly forbidden except for students enrolled in CS-350.
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