CompSci 102 Discrete Mathematics for CSSpring 2005 Forbes HW 3Due Monday, March 28Please note the following on your assignment: (1) all people with whom you collaborated,(2) resources you used other than the book and class notes, and (3) how much time you spentworking on the assignment.Submit using assignment name hw03.1. (36 pts.) Book Problems(a) Section 4.3, Exercise 40(b) Section 5.1, Exercise 38(c) Section 5.2, Exercise 40(d) Section 5.3, Exercise 342. (10+5 pts.) Monty HallConsider a variant of the Monty Hall problem in which there are n doors rather than theusual three. Once again, there is a prize behind one of the doors, and a goat behind each ofthe other n − 1. After the contestant choos es a door, the assistant opens one of the unchosendoors to reveal a goat. Under the switching strategy, the contestant then picks one of then − 2 remaining doors with equal probability.(a) What is the probability of winning under the switching strategy (as a function of n)?(b) [Extra credit]Suppose there are n doors and m ≤ n − 2 prizes (each one behind a different door).Again, the contestant picks a door and the assistant then opens a goat door. (Thecondition m ≤ n − 2 ensures that this is always possible.) What is the probability ofwinning under the “sticking” strategy? What is the probability of winning under the“switching” strategy?13. (20 pts.) A paradox in conditional probability?Here is some on-time arrival data for two airlines, A and B, into the airports of San Diegoand Chicago. (Predictably, both airlines perform better in San Diego, which is subject to lessflight congestion and less bad weather.)Airline A Airline B# flights # on time #flights # on timeSan Diego 500 453 200 188Chicago 300 211 900 685(a) Which of the two airlines has a better chance of arriving on time into San Diego? Whatabout Chicago?(b) Which of the two airlines has a better chance of arriving on time overall?(c) Do the results of parts (a) and (b) surprise you? Explain the apparent paradox, andinterpret it in terms of conditional probabilities.4. (20 pts.) Happy families(a) Consider a collection of families, each of which has exactly two children. Each of the fourpossible combinations of boys and girls, bg, gb, bb, gg, occurs with the same frequency. Afamily is chosen uniformly at random, and we are told that it contains at least one boy.What is the (conditional) probability that the other child is a boy? Justify your answerwith a precise calculation.(b) For the same collection of families as in part (a), suppose now that we pick a childuniformly at random, and are told that the child is a boy. What is the (conditional)probability that the other child in this family is a boy? Justify your answer with aprecise calculation.(c) Explain the difference in the answers to parts (a) and (b) with reference to the twosample spaces.5. (14 pts.) IndependenceThis problem illustrates that events may be pairwise independent but not mutually indepen-dent. Two fair dice are thrown. Consider the follow ing three events:A = the first die is odd;B = the second die is odd;C = the sum of the dice is odd.(a) Show by directly counting sample points that events A, B are independent; that B, Care independent; and that A, C are independent.(b) Show that the three events are not mutually independent.CompSci 102, Spring 2005, HW 3
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