HARVARD MATH 152 - Homework Assignment #9

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MATHEMATICS 152, FALL 2003METHODS OF DISCRETE MATHEMATICSHomework Assignment #9Due: November 25, 2003Required Problems1. Apostol, Section 13.4, exercise 1. If no simpler approach occurs to you,try the following:To show that sets X and Y are disjoint, show that• if x is in X it is not in Y• and if x is in Y it is not in X.To show that sets X and Y are equal, show that• if x is in X it is also in Y• and if x is in Y it is also in X.2. Apostol, Section 13.4, number 4. To prove the formula by induction,• first show that it is true for n = 2,• then assume that it is true for arbitrary n and show that it is truefor n + 1.3. Let the universal set S be the set of all 10000 undergraduates at a uni-versity. Here are some subsets.• A is the set of 2500 freshmen• B is the set of 3000 athletes• C is the set of 5000 women• D is the set of 200 football playersThere are 1200 female athletes, but none of them play football. Of thefreshmen, 1000 are athletes and 1250 are women. 400 are both.Express each of the following subsets in terms of A, B, and C, andspecify its size.(a) The set of female upperclassmen.(b) The set of male freshman who are not athletes.(c) Translate into the language of sets the statement, “If a freshmanplays football, then he is a male athlete.”1(d) An alumnus asks the university to choose a student at random toreceive a scholarship. What is the probability that the recipient iseither a male freshman athlete or an upp erclass woman?4. Apostol, section 13.9, exercise 10.5. Apostol, Section 13.11, #7 and #8. In part c, you need to know thatace, 2, 3, 4, 5 in a suit counts as a straight flush.6. (This problem is loosely based on some work with a recently-discovered1774 census of Rhode Island)A genealogist, on analyzing names in 18th-century Rho de Island, hasascertained the following:• The probability that a male child was a slave was 0.3.• Male children who were slaves were given classical names like “Cae-sar” and “Aesop” with probability 0.6.• Male children who were free were given classical names like “Caesar”and “Aesop” with probability 0.2.Event A is “the child was a slave” and event B was “the child had aclassical name.”(a) Using set-theoretic notation, express the event “the child was ei-ther free or had a classical name” in terms of events A and B andcalculate its probability.(b) Are events A and B independent? Justify your answer.(c) The genealogist encounters the classical name “Cicero Greene”.What is the conditional probability, given his name, that Cicerowas a slave?(d) The genealogist learns that Cicero had one sibling, his brotherRoger. Either both were slaves, or both were free. What is theconditional probability that b oth were slaves?7. In an eight-team football league there are four referees. At the start ofthe season coins are distributed to them at random. Three of the coinsare fair ones, but the fourth has two heads. Event A is that referee Rreceives the two-headed coin. An astute coach notices that the first threecoin flips of referee R have all come up heads. This is event B. The coachproposes to say to the referee, ”Are you using a two-headed coin?” buthe wants to know the probability of Event A in order to be sure that hehas a good chance of being correct.Calculate the conditional probability, given event B, that referee R isusing the two-headed coin.28. (a) Determine how many ways there are to select a subset of 5 of the13 spades in a deck of cards.You may leave your answer in terms ofproducts of integers, but expand any factorials or binomial coeffi-cients.(b) Determine how many distinct bridge hands contain 5 spades, 3hearts, 3 diamonds, and 2 clubs. As above, you may leave youranswer in terms of products of integers.(c) Determine how many distinct bridge hands have 5 cards in thelongest suit, 2 cards in the shortest suit, and 3 cards in the othertwo suits.9. The Queen of Sheba has come to Jerusalem to find a prophet. There aretwo sorts of prophets: true prophets, who speak the truth nine times outof ten, and false prophets, who speak the truth half the time. Prophetagents are forbidden to reveal explicitly w hich of their prophets are trueones.The queen wants to be more than 90% certain that the prophet sheselects is a true one. She hires a prophet agent who brings out threeprophets: two true ones and a false one. “2 out of 3 – that’s not goodenough, is it?” the agent asks. “Sure it is,” says the queen, “as long asI can ask one yes-no question.” “Ask away,” says the agent.The queen asks prophet 2, “Is prophet 3 a true prophet?” On hearingthe answer she makes her selection and heads home. How did she do it?Hint: Event B is “answer is yes”, B’ is “answer is no.” Events A1, A2,A3 are respectively ”prophet 1 is a true prophet,” etc.The queen must be sure that for either answer, her selection (made afterhearing the answer) satisfies P (A|B) (or P (A|B0)) > .9.For a more fanciful version of this problem, see #3 in the Bayesian Bible.10. In the admissions office at Monty Hall University there are four inter-viewers. Three of them, Fl, F2, and F3, are friendly, while the fourth,U, is unfriendly. Every morning the Dean of Admissions assigns themrandomly to offices 1, 2, 3, and 4, with an equal probability for eachpossible assignment. A student arrives for an interview and is asked toselect which office he wants to be interviewed in. He chooses office 1and learns that the interviewer in there is busy for the next half hour.“While you are waiting,” says the Dean to the student, “I would like youto meet one of our friendly interviewers. From offices 2, 3, and 4, I willchoose the lowest-numbered friendly interviewer.” He opens the door ofoffice 2 and introduces the student to an interviewer.This is Event B – the lowest-numbered available friendly interviewer wasin office 2.Event A is that office 1 contains the unfriendly interviewer.3(a) Enumerate all the ways of assigning interviewers to offices that leadto Event B . Assign a probability to each, and show that the sum ofthese probabilities equals the probability of Event B.(b) Given that Event B has occurred, determine the conditional prob-ability of event A.(c) What is the probability that the friendly interviewer to whom thestudent was introduc ed by the Dean was F1?Exploratory Problems1. Show that the permutationsa = (15)(24)(36)andb = (14)(26)(35)generate a subgroup of S6that is isomorphic to


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