feb25 - Illinois Institute of Technology Department of...

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Unformatted text preview: Illinois Institute of Technology Department of Computer Science Lecture 10: February 25, 2008 CS 330 Discrete Structures Spring Semester, 2008 1 Probability When the weather-woman says there is a 30% chance of rain, what does she mean? Does she mean that: rain will fall on 30% of the viewing area? in the last 100 years, it rained 30 times on this date? under present conditions, recorded history for comparable conditions shows it rained 30% of the time? Similarly, what does it mean for an algorithm to be correct 99% of the time? Can we trust it to guard our nuclear warheads? In order to answer these questions, we first need to know a little about probability . 2 Basics of Probability Consider the following amoeba-like graph: z } | { T o t a l i t y o f e v e r y t h i n g t h a t c o u l d p o s s i b l y h a p p e n z }| { some possible event z } | { S u b s e t d e fi n e d b y a n e v e n t o r g r o u p o f e v e n t s S We wish to measure the likelihood of event S occurring. We call this Probability of S and we write it as Pr { S } or P [ S ] . Consider the following experiment: We will flip a fair coin. We have the following universe of outcomes: CS 330Spring, 2008 2 Lecture 10: February 25, 2008 H E A D S T A I L S Our intuition tells us that Pr { TAILS } is 50% or 1 2 because TAILS happens about half the time. Consider this experiment: We will roll a fair die. We have the following universe of outcomes: Our intuition tells us that Pr { Rolling a one } is 1 6 , because rolling a one occurs about one sixth of the time. From these we can see a simple definition of Probability as: Pr { S } = number of events in S number of events altogether This definition will be ample for our purposes....
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feb25 - Illinois Institute of Technology Department of...

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