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Unformatted text preview: University of Illinois Fall 2009 ECE 313: Problem Set 5 Conditional Probability, Law of Total Probability, Bayes Formula Due: Wednesday September 30 at 4 p.m. Reading: Ross Chapter 3; Powerpoint Lecture Slides, Sets 914 Noncredit Exercises: Chapter 3: Problems 1, 2, 5, 10, 12, 16, 31, 38, 39, 51, 52 Theoretical Exercises 1, 2, 8, 16; SelfTest Problems 114. Reminders: No class on Wednesday September 30 and Friday October 2 but office hours as usual next week 1. [Picking a random subset] The experiment consists of drawing a subset of size k at random from a set of n distinct objects. (a) How many different subsets of size k are there of the set of n objects? What is the probability of drawing a specific subset, say { 3 , 7 , 23 ... } of size k ? (b) Now suppose that we draw objects one at a time without replacement until k objects have been drawn. Let A 1 , A 2 , ..., A k denote the events that the first, second, ... kth draw respectively gives us some element of the specific subset of interest. Note that we get our desired subset if and only if the event A 1 A 2 A k occurs. What is P ( A 1 )? What is P ( A 2  A 1 )? What is P ( A 3  A 1 A 2 )? More generally, what is P ( A i  A 1 A 2 A i 1 )? Use these results to calculate P ( A 1 A 2 A k ) and compare to the result of part (a). 2. [The beginning of spring training] A baseball pitchers repertoire is limited to fastballs (event F ), curve balls (event C ) and sliders (event S ). It is known that P ( C ) = 2 P ( F ). Let H denote the event that the batter hits the ball. Whether H occurs or not depends on the pitch, and it is known that P ( H  F ) = 2 / 5 ,P ( H  C ) = 1 / 4, and P ( H  S ) = 1 / 6. If P ( H ) = 1 / 4, what is P ( C )? 3. [Exercises in Conditional Probability] This problem on conditional probability has three unrelated parts: (a) If P ( A  B ) = 0 . 3 ,P ( A c  B c ) = 0 . 4, and P ( B ) = 0 . 7, find P ( A  B c ) ,P ( A ), and P ( B  A )....
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