Final Exam

Final Exam - MATH 30 (Spring 2801 Lecture 3) iiistructor:...

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Unformatted text preview: MATH 30 (Spring 2801 Lecture 3) iiistructor: Roberto Schoiimanr: Final Exam Last Name: First and Middle Names: Signature: UCLA id number (if you are an extension student, say so): Circle the discussion section in which you are enrolled: 1A (T 9, Matthew Keegah) 18 (R 9} Matthew Keegae) 10 (T 9, Yin Wang) 1D (R 9! You Wang) 1133 (T 9, Christopher McKiniay) 217' (R 9, Christopher McKinisy) When the instmctions to a. question ask you to explain your answer: you shoulé show your work and explain what you are doing carefully; this is then more important than just finding the right answer. F'ieatsea write olesriy and make clear What your solution and answer to each pzoblem is. When you continue or: another page indicate this Clearly. To caecei anything from your solution, erase it or cross it out. You are not allowed to sit dose to students with whom you have studied for this exam, or to your friends. Enjoy the exam, and Good Luck 3 | Tot s1 I I! l-'-.-- '2) (10 points) In a group 0f students, 8 are freshmen 21.1103 8 are sophomores. in how many ways can the stuéents in this group form a waiting line: if freshmen and sophomores shouéd alternate in the line? 0 need to compute facteriais, pets-“ereg pet‘n‘lutetions and combinations.) (Ne explanatioe needed, gust the answer is enough.) 4. i; a x: g, A f} s fie 2* - _.! m; - $ 7- g g? "W “I? é-‘ug WM «2“» “gr/“(k “is W w «a “2 a a ‘ (“'3 M: mi». a! a- : g; m 2 3 M , "w! ’33 i’\ z. s my 2 5.40%.? 2) (10 points) ROB a die 2 times. What. is the conéitisnai probability that the maximum of the faces shown. is 6, given that the miminmm of the faces shown is 4‘? (Give answer as a fraction. or in decimai form.) (Show and expiain you}: work.) 5 i Wm K3 . ,3 = “aw § x - 1*;- :9»: a , ‘34; 2‘ I *v r (f 5w - K . {3‘3'PZ4‘ iami'KVlzvfififijféfb “3.x “E ‘3 fix ~- g <s« 6 ~ f” . g g. 34"“ X) ‘r W “y; “MMm—wamm n «w w. “WNWMHWW 5 , g x} 3) (10 points) You are deait 8 cards item a. standard deck of 52 cards. Com— pote the probab'fiity that among those cards you have: 2 cards of one denom— ination, 2 cards of another denomination, 3 cards of a third denomination, and one singie card of a fOIch denomination. (No need to compute factoriw ale, powers, permutatimls and combinations.) {No explanation needed, just the answer is enough.) 4) (10 pomts) ROI] 9. fair die. Consider the events A m {1:3}; 8 m {3,4}. Are A ami B independent? (Expiain your answer carefully.) .m, 113" 5:; $2 ’5‘ is :5; w. j M f i f é \ , M ,3 WW 3% as 2:; y f a "’ gfiv £8.85- v,“ i L w‘ ya w»- : "W (if) perms) A group of people has '16 adelte 2ian 20 children; One of the adults is called Ann and another one Beb. We want to seéeet 2 aéulrs and 8 children tie ge 0:1 a trip. They will go in a vehicle with room for 30, and one ef the selected eduits has to be the driver. (All éhe adult-s ere assumed to be peéeetiel drivers.) Suppose that every pessibie choice of driver and passengers is equally éikely. What is the probability that Arm will be the driver er Bob will he a passenger (110% the driver)? (Provide numerical answer as a. fraction or in cleeimel form.) (Show ané explain your work.) 6) (10 paints) A screening test for a disease shows a false positive with probability 1% and a false negative with probability 2%. in the population 40% of people have that disease. Given that someone tested positive for the disease? what is the probabiiity that lie/she has the disease? (Proviée a numerical answer in decimal form? or as a percentage.) (Show and explain your work.) 5,, gags“, WW.» E '7) (10 poin’és) A box ccmtains 2 fair coins and 1 coin with two €33.18. One coin is selected at random from {this box and flipped 5 times. Let X be the random. variahie that gives the number of heads in the .5 fiips of the coin. Compute P(X 2 4). (Provide a. numericai answer as a Emotion or “in decimal form.) (No explanation neeéedle just the answer is enough.) -N:~ 8) (10 points) Suppose that. {be independent evenis A. and. B have probabil— ities 13(14) :: .8 and [3(8) m ,5. Compute PTA U B (Provide a numerical answer in decimal form.) (Show and expiain your work) " W C m if? d 2” m ‘ E 2» M? a 3L 2‘ F E [R «’6 EV? m 4 a, W , “3" xx 5. a w 1' w z,- L“; v“ f“. w a! ” “‘— $ ff 5 f' 9) (10 points) A friend of yours verified to wriépe a compete: code to compute the variance of a disereée mmle variabie X ‘ She knew {she correct. formuia: Var(X) x Enha— ~ Mgszfiz.) But, by mistake she lefi 1{he square out Of 1{he formula; maiiing the computer compute instead $7102, — ;¢X)pX(n). What numerical answer did the cemgmter give her? (Expiein your answer eerefuliy.) i0) (10 peints) A fair coin is flipped 3 times. Let X be the random variable that gives the number of heads shown. in the first two flips of the coin, and let Y be the random variable that gives the number of heads shown in the last two flips of the coin. (For instance, fer the euteeme hht we have X m 2} Y m 1,, and fer the outceme tth we have X = G? Y m 1.) Provide a table that gives the joint distribution of X and Y, Le, a. table that gives the numbers P(X w» k3,}f = l)‘ k m 091,2, (7 = 0:1,2. (The numbers in the table should be given as .fi‘eetions, or in decimal form.) (Show 21ml explain year work.) s e! 5;: 5“: s i V; 1’ A ’5 § 1! v f «m we ,z gaff/E f fa €me %*M:> 11) (10 points) A {:Grztinuous random variable X has probability densiicy fuiicéion given below, where C is an appropriate number 0 if 5<0, 13(5) = Us if @354, 6’3""3 15521. Find ilie value. of C anti (301121311556 the mean ii}; of X. (Provide numerical answers in decimal form‘) (Silow and explain your work.) 12) (10 points) A continuous random variable Y has probabiiity density func« tioa given beiow. 0 if s<»—10rs>1, fy(5) = 1+8 if WE§S<G; Ems if 03.331. Compute Var(Y). (Provide a numerical answer in (1601111211 form.) (Show and explain your work.) ........ .. ,. V 5. é " (P. X 5% w a: z X“ w ; a M w x“? 59% a § f'xwxz W w 7’ 3; 3 g“ é“ n2 a.» «'2 M” a f é . ,5 a o mm ., é z" _ g > : «xi 3 a»; E ‘ wwffi ; mm E3) (10 points) Suppase that W is a random variabée with normal distribution wifih mean m1 and stanciamd deviation 2. (301211312116 P(m3 < W < 0). (Pravide a numerical answer in decimal form.) (Show and explain your work.) r 1 Bi} a :5 q J w km,” 7, L, M Kw} ,, : f», rid ‘ f M . "\ : M “i fix » m «6' g {or g :2: W “w 51? “W M: M am <23»! j; 2;; x“ MM, fw— "‘8 WW - r y 2 5 g” (2%:— d; f z “x “W Men- mflw ? 14) (10 points) You have an egjpoinmleu: with a friend at. 4:00 p211. You know from experience that your friemi wiil eryive at a. time that diKers fzcom 4:00pm by a random amount. T With normed distribution with mean 0 and siendard deviafigion 5 minutes. (A negative value of T means that your friemi arrives early, a. positive value of T means that. your friené arrives late.) Given that. your friend has 110%: arrived yet e’é 6%:05; whet is the eonditionai probebfiity that he WEB not yet have arrived 4:10? (Provide a numeyioei answer in decimal form.) (Show and expiein your work.) 15) (10 poin‘zs) Toss a fair coin 200 times. Use the central limit theorem to finé an approximmtion for the probability that the number of heads is at least 60. (Provide a numerical answer in decimal} form.) (Show and explain your work.) 8 fl Tabie ofthe Standaré Nermai Diswibution TABLE 0? THE STANDARD NORMAL DISTRlBUTiON Areas mader the Standard Normai Curve from «so to 5 (see Figuye 8.1). 1"? _).¢—9—"< x 4 Figure 8.1 919 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 9 .5359 .5754 .6141 .6517 .6879 .7224 .7549 .7852 .8133 .8389 .8621 .8830 .9015 .9177 .9319 .9441 .9545 .9633 .9706 .9767 .9817 .9857 .9890 .9916 .9936 .9952 .9964 .9974 .9981 0 I 2 3 4 5 6 7 8 00 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 5319 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7258 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7518 .7580 .7612 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7881 .7910 .7939 .7967 .7996 .8023 .8051 .8078 .8106 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9636 .9625 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9953 .9955 .9956 19957 .9959 .9960 .9961 .9962 .9963 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 29 .9986 ...
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This test prep was uploaded on 04/03/2008 for the course MATH 3C taught by Professor Schonmann during the Spring '07 term at UCLA.

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Final Exam - MATH 30 (Spring 2801 Lecture 3) iiistructor:...

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