201midterm1-sol - SIMON FRASER UNIVERSlTY DEPARTMENT OF...

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Unformatted text preview: SIMON FRASER UNIVERSlTY DEPARTMENT OF MATHEMATECS Midterm 1 MACM 201 Spring 2013 Instructor: Dr. L. Stacho February 6, 2013, 12:30 —— 1:20 pm. Name: (please print) family name given name SFU ID: student number SFUwemail' Signature:- Instructions: . Do not open this booklet until told to do so. . Write your name above in block letters. Write your SFU student number and email ID on the line pro vided for it. . Write your answer in the space provided below the question. if additional space is needed then use the back of the previous page. Justify your answers and make the method you are using clear in every case unless it is explicitly stated that no explanation is needed. . This exam has 3 questions on 5 pages (not includ ing this cover page). Once the exam begins please check to make sure your exam is complete. . N0 calculators, books, papers, or electronic devices shall be within the reach of a student during the examination. . During the examination, communicating with, or deliberately exposing written papers to the view of, other examinees is forbidden. i- --- MACM 201 Page I 0f5 1. Discrete Proba bility [4] (a) Ten tickets labeled 1 to 10 are placed in a box. Three of these tickets are then 7 drawn, in succession and without replacement, from the box. i) Find the sample space for this experiment, ii) Find the probability that the labels on the drawn tickets are increasing, i.e. the label on the first ticket is smaller than the label on the second and the label on the second ticket is smaller than the label on the third. Justify your answer, define events and show formulas you use. i) P: { (use): $.33 e 13‘" “’3‘ x #2; .x H‘. 3”} ti) let .~. lD-‘s-S’ {no '1 b 51- “ ‘° “stung [thermal-21+! .—. 36 , c = Z 2}, . « .. Kc“ ZZ.‘ =Slh¥lb *‘ K .‘=?’J:H1 [Oi-6+ 3+l 61‘1““ xqjtrft =?«6+"*‘=28 s.}. 3 .120 ‘ :=33=i~. g ‘ - M133.“ ( y _ l Al ‘20 ___ l . if nu Hi? P‘R’figag'gi‘ ._ . \ ._. --" = X..Q. :zf5=i!-I l fi" 1 to t = -- fl :10 KS: tag-$6! q "s" . [4] (b) A fair coin is tossed three times. What is the probability of having three heads if the first toss re5u|ts in a head? A'. cum-k -\l~L\- “Were MAC at: o'b‘l'athncok Q:<u¢nk Mk- RISE ‘l‘b$s is 0L. ”Raft ' Ant) L gram . ii...” = 2.. = l ?r (3) g: :1... | MA CM 201 - ' _ Page 2 of 5 [4] (c) A fair coin is tossed five times. If: X = X1 - X2, where X1 counts the number of heads and X2 counts the number of tails that result. i) Determine the probabiiity distributiorimc X1,X2, and X. ii) Determine the expected value of X. Show your work. i) K‘; fags‘ngz values OHS‘ k. We 3' oskss, arx.ms(f).(g.i-(gi . (Jig) X , pie-$31 values: 0» S E. 311; ,_ ‘ " ~. arm..k7=(D-(%)-(t we oskét Kay-x1. wait. ”due; 41-3 H ,\’> f tho-s). Mxlso Axis) ..R(xl_.o-\ “(Kg—“0:?rhk‘d A XML-EH; \) ' Mxrifi. ’i’rm—z A R1=1):?E(>g‘=9_) “r: (x... a . m (xln A M2) .. a. (3.1) V? (Kz'5): ’B‘CKfi-L‘ '\ KL: 0.— ?l (Kt-f“) ‘ ?Y(Kfi'§ ’\ XL:DU:?§(KL: 8-)7 ?£ (K: 5—3 «- ' ' \e. Rik “j Kt «not XL are; Slmwxml famo‘owx wands _ 3 y . twee €00 ,. E (m) = .2. Ba “Mali-ha 0(— CK?ZCJWLHQUK I E(K\ = E( “”KZ}: z 600 - E09) :- Evgso, 12,...“ MAOM 201 . . ' ' - Page 3 Off)" 2. Inclusion/Exclusion N = N ZNRQ) +2 N(CiCj)A“"‘+(——1)tN(6102...Ct), 1<i<t 1<i<j<t 8k; : Z N(ci1ci2“Cik-):1§kgta 1Si1<i2<m<ikgt m+1 m+2 ,m t sm( 1 )sm+1+( 2 )sm+2_...+(1it (Wm Lm : sm—( m )Sm+1+<m+l)3m+2--.+(—1)z'm(t1)8t,lgmgt. mil —1 $3 | I [4] (a) How many positive integers m not exceeding 99999 are there, where o the i—th digit from the right is‘ at most 2’ for all i, and o the sum of all digits is 8? For example 50111. 50021 or 4301 are such integers but 50030 or 21 are not. " Kex‘rxwxsxzxi 05K;$°l (K (V‘- 54‘2- 1°) is 'ch number n .W'\: 24-5 K * “L ‘ ‘3 same; \ :h-S) . X 01c integer Sob’noms ’r» x‘ + x1 1: x3 + q (my) \a c; 5.2. reed» *c a i“ N 3 (S'+?'i)=(t81) mcngt‘z \ M (CM Mama) .Mcm e.) mm (.3) “(ask A“ ail-Met. mes are. 0. WM “(563; Esau—rd: (2) ”(5W E1431 (6)] [(3) WM) 25—) NM (fl) waJ‘ngJi) (9) 14(62ch (“lutgcastfl MA CM 201 ' Page 4 of 5 [4] (b) You have just got the brand new Intel 7—core processor LOVEMATH. For a test you prepared 10 distinct tasks that you want to run on these 7 cores. For now, you do not want to overload the processor, so you decide to run these tasks on exactly 4 of the 7 cores. ln how many ways you can distribute these 10 tasks? Assume the 7 cores are not identical. . For aha om—‘l‘auonc. assignme-Alt oh l0 kaslcs 'l'D 3‘ (MES lt'l‘ C- (C:\.- 3% lac, vaPer'kE Cart :1: [S Viol.— usgpfi in +c§+ we; (mu-3r) 9m=l32\0r~m3‘° 3, fl. °‘l" wags £61” -lrlM:. \‘cb'l‘ ls EguS3“llllS‘-1+(§)S[“l§lgc+l$)g} i [4] (c) Jen, Mike, Lora, and Michelle bought one of each of the following fruits: apple, banana, strawberry, kiwi, orange, and pear. In how many ways they can eat each exactly one fruit assuming Jen does not eat apple and banana, Mike does not eat apple, banana and strawberry, Lora does not eat kiwi, orange and pear, and Michelle does not eat kiwi and orange. Use the method of rook polynomials to answer this question. 1 ”2.. rlcam): lltyfifil“) 1' 1 \-\'\O}g #3314th *l6 [1 a 9k “Oil-S4“ . “PVC aka “Jib—ORE 1%l8hmlc PCFSOUNQ leak Q. _ Propzr‘qs ‘FC(SM L' ass‘igherl Gail: lhea, glo m3? teak _ l0--( Mrs) + 63014) PHD-('3) + \6 J: T M519 4* D’l’ wotag Data can eacl— llwt Gui-lg. MACM 201 - Page 5 of5 3. Generating functions [4] (3) Oscar wants to split 19 identical bookmarks to three of his friends Mark, Susan, and Dmitri. Is this possible if he wants to give an even number of bookmarks to Mark but no more than 9, a multiple of three bookmarks to Susan but no more than 12, a multiple of five bookmarks to Dmitri but no more than 15, and he wants to give at least one bookmark to each? Set up the generating function and explain why Oscar can or cannot split his bookmarks this way (you do not need to simplify the generating function). - new: a?» ‘2‘ «r Kc «- KB) = lie-.04 Dmilrci - (XY+ xi“) 1- Km) a {15(3) . kcnzralfi was :uuxclrhux POO} Janina] . £3097 ‘ fad“) - \a . .9; «Jr mega; Sec Oscar is line. CoeGCac'vavxlr 9‘“ K m {‘08)- wlflicbx Com be. Sccu. is Q... l‘kthce, DSCOLI' Can SPli‘l‘ Lmkw‘q‘l‘S. [4] (b) Find the coefficient of 2:20 in %. You do not need to simplify the expression. '5- - I ”3 Lime. )4 l :2: gm X+LHL ’ ’75.... + (m? and” (w? (rind5 2 ‘4 2| ' Con-C-ml” x“ m ‘ ts (”‘3 ltsl "l”l (ii-x13 l .coeoc, 01, x” in {300 is ”llll" Lll'l‘q) ...
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