PS05 - University of Illinois Fall 2009 ECE 313: Problem...

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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 9-14 Noncredit Exercises: Chapter 3: Problems 1, 2, 5, 10, 12, 16, 31, 38, 39, 51, 52 Theoretical Exercises 1, 2, 8, 16; Self-Test Problems 1-14. 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, ... k-th 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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PS05 - University of Illinois Fall 2009 ECE 313: Problem...

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