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Unformatted text preview: IE 300 / GE 331 Spring 2009 Homework #1 Due: Feb. 5th Problem 1. Let A and B be two sets. (a) Show that A c = ( A c ∩ B ) ∪ ( A c ∩ B c ) , B c = ( A ∩ B c ) ∪ ( A c ∩ B c ) . (b) Show that ( A ∩ B ) c = ( A c ∩ B ) ∪ ( A c ∩ B c ) ∪ ( A ∩ B c ) . (c) Consider rolling a fair sixsided die. Let A be the set of outcomes where the roll is an odd number. Let B be the set of outcomes where the roll is less than 4. Calculate the sets on both sides of the equality in part (b), and verify that the equality holds. Problem 2. Out of the students in a class, 60% are geniuses, 70% love chocolate, and 40% fall into both categories. Determine the probability that a randomly selected student is neither a genius nor a chocolate lover. Problem 3. We roll two fair 6sided dice. Each one of the 36 possible outcomes is assumed to be equally likely. (a) Find the probability that doubles are rolled. (b) Given that the roll results in a sum of 4 or less, find the conditional probability that doubles are rolled. (c) Find the probability that at least one die roll is a 6. (d) Given that the two dice land on different numbers, find the conditional probability that at least one die roll is a 6. 1 IE 300 / GE 331: Homework #1 2 Problem 4. We are given three coins: one has heads in both faces, the second has tails in both faces, and the third has a head in one face and a tail in the other. We choose a coin at random, toss it, and the result is heads. What is the probability that the opposite face is tails? Problem 5. Urn experiment: An urn contains 5 red, 12 blue and 4 white balls. (a) 2 balls are drawn from the urn without replacement. What is the probability of drawing two white balls? What is the probability of drawing 1 red and 1 blue ball? (b) 2 balls are drawn from the urn with replacement. What is the probability of drawing two white balls? What is the probability of drawing 1 red and 1 blue ball? Problem 6. Seventeen dogs are in a shelter. Eight of the dogs are black, seven of the dogs have short tails, and twelve of the dogs have long hair. There is only one black dog that has a short tail and long hair. Four of the dogs have both short tails and long hair but are not black. Two of the black dogs with short tails do not have long hair. If all of the dogs in the shelter have at least one of the mentioned characteristics, how many black dogs have long hair but do not have short tails? (Hint: use a Venn diagram) Problem 7. How many 4 letter words can be made using letters from CALCULATE when: (a) all letters are different. (b) two ’a’s and another identical letter pair is used. (c) two identical letter pairs are used. (d) one pair of identical letters and 2 other different letters are used. 1 / 4 IE 300 / GE 331 Homework #1 Solutions Problem 1....
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This note was uploaded on 09/08/2009 for the course GE 331 taught by Professor Negarkayavash during the Spring '09 term at University of Illinois at Urbana–Champaign.
 Spring '09
 NegarKayavash

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