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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 ,
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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 adults 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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 Spring '07
 SCHONMANN
 decimal form, numerical answer

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