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Unformatted text preview: MATH 3C (Spring 2007, Lecture 2)
Instructor: Roberto Schenrnenn
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: 2A (T 10F Akemi Kashiwada) 2B (R 19; Akemi Kashiwarla)
2C (T E0, Wenhua Geo) 29 (R 10, Weuhua. Geo)
213 (T 10, llhwan Jo) 2F (R El}, lllzwan $50) When the instructions to a question ask you to explain your zusswer7 yeu
should show your work and explain whet you are doing carefully; this is then
more important than just ﬁnéing the right answer. Please, write clearly and
make Clear what your solution and answer to each problem is. When you
continue on another page indicate this clearly. To cancel anything from your
solution, erase it or cross it out. You are not allowed to sit Close to students
With Whom you have studied for this exam, or to your friencis. Enjoy the exam, and Good Luck 1 1) (10 points) Roll :1 die 2 times. What. is 1ighe condiijmzal probability that:
the sum of the faces shown is; 6? given Ehat at least one of the faces shown is
a 4? (Give answer as a. fracéion or in decimal form.) (Show and explain your
wark.) .w’“ 1K"
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'53 2) (10 peims) In how many ways: can we give 3 prizes to a gmup of 190
people; if each prize can oniy be given to (me person; and each person can
receive a? most. one prize. (vaide a. numerical answer.) (No expianation
needed, 312m; the answer is enough.) 3) (E0 points) You are dealt 6 cards from a standard deck of 52 cards. Con:
pute the probabiiity tho: among; those cards you have: 3 cards of one denomi—
nation and 3 cards of another denomination. (No need to compute factorials,
powers, permutations and combinations.) (No explanation needed? just. the
answer is eaough.) 4) ($0 points) R01} 3. fair die. Consiéer the events A = {1; 3}, B m {3,4, Are A and B independent? (Explain your answer carefuily.) {‘x g‘ 3;: my» 3?} w wm E 2 ’ 2 4‘
W/ / Km 1;: Wt. . V . .4  .p a w « x,
«x 1 w .v 5) (10 points) A greup of people has '20 adults and 20 children. One of {he
adults is called Ann and another one Bob. We want te select 3 adults and '7
Children ﬁo go on a. trip. They will go in a vehicle with room for £0, and one
of the selected adults has te be the driver. (All {he adults are assumed to be
potential drivers). Suppose that any possible choice of driver and passengers
is equally likely. What. is the probebilitv that Ann will he the driver and
Bob will be a passenger (not the driverﬁ (Provide a numerical answez es a
fractioa or in decimal farm.) (Show and explain your work.) "28%.
“a. 6) (10 points) A screening test for a. disease shows a. false positive with
probability 1% and a false negative with probabiiity 2%. In the population
03% of people have thee disease. Given. that someone tested positive for
the disease, what is $118 probability that heshe has the disease? (Provide a
numerical answer as a. fraction or in decimal form.) (Show and expiain your
work.) 7’) (10 points) You have three boxes. The ﬁrst box contains 200 blue bails and
100 yeiiow baiis. The second box contains 200 blue belie and 200 yeilow balis.
The shird box oon’sains 200 biiie bells aged 300 yeliow bails. A box is seieoted
at random; and from this box balls are selected Without replaoeniens. Let.
S be the random variable ihet gives the nurnﬁer of yellow bails among {he
selected ones. Find P(S§1). (No need to compute feeﬁoriais, powers,
permutations and combinations) (No explanation needed7 just the answer
is enough.) 8) (10 points) Suppose them: the independent events A and B have probabil—
ities PUD m .6 and. PUB) m Compute PTA U B). (Provide a numerical
answer in decimal} form.) (Show and expiain your work.) m wig/f Wm
iv; a 
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s,
f :7
g 03‘ ﬂ, :q a"
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W» .ﬁ Z X In; ﬂ 9) (10 points) The mean deviation of a discrete raziéom variable X is deﬁned
as Zn m th grain)a Where pxm) : P(X m n) is the probability mass function of X 7, and the sum is over all the values that X can take. Compute
the mean deviation 01" a random variable X that has binomial distribution
with 4 attempts and probability E /2 of success each time; ﬁe, X N BM; 1 /2).
(Give answer as a fraction or in decimal form.) (Show and explain your work.) 10) (10 points) Suppase that X and Y are independent random variables with
pmbabiiity 11121.55 funciiions given by: pX(—1) = .3, pg; (0) mm: .4, pX(1) m .2,
MO) = ,12 and pyﬁ) m .2, py(2) m .8. Compute P(X ~i— Y 2 2). (Give
answer in (iecimal form.) (Show and explain your work.) ,, 5 fax g
{jgxgysi 33$ “35w
e A w _ m ‘ ’x 5‘ v; EJfli' e M" 5:» W 3 w‘ ‘
.M w ‘3 F
‘u w r 32; z
~ 1/ w = ¢~._
w (5 gum " vi 1.4 ii) (10 points) When Helen tosses a. dart at a target, the (Eistance (in inches)
from the center of the target that she hits is we]? éescribed as a. continuous
random variable X with probability densityr functien given below, where C’
is an appropriate number. 0 if s<0 ors>§0,
XS ‘” 05 if ogsgie. 21) Find the value of C. b) If she tosses the dart at the target 5 times, what is the probability that at
least once the dart hits the target less than 2 inches from the center? Assume
independence of the 5 tosses. (Provide numerical answers in éecimel form.) (Show and explain your work.) 12) (10 points) A cantinuous random variable Y has probability density func—
tien given below. 0 if s<——1ors>l.
£43 = 1+5 if—1§5<:Q
lws if OSSSl. Compute VarCY). (Provide a numerical answer in decimal form.) (Show and
{explain your work.) éggﬁ} é :2
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gig/19¢ Véq/ (y; a: f 26% {é} “ii” “" f f, I"; W“ M 1.? 1w :“‘”~’&5§f“
m >’ g j 13) (10 points) Suppose mat. W is a yandom variable with normal distribution
with mean ——1 and stanéard deviation 2. Compute P(—3 < W < 415). (Provide a. Immez‘ical answer in decimal form.) (Show and explain your work.) L t "
mm.
M.
Kim
mm, {Ha 14) (10 points) A certain substance in the blood is distributed as a normal
random vari able with mean 290 and standard deviation 10 (when measured
in rng/dl). People who have an amount larger than 225 nag/d} are diagnosed
as having a certain medical condition. When the amount is larger than 215
mg/ (11, doctors call the patient (If the amount is between 215 ng/ CH and
225 mg/dl they recommend life style changes, to prevent the condition from
developing; if the amount is larger than 225 rug/(ll they start treating the
condition with medications; but if the amount is less than 215 :ng/ (:11 they
do not even call the patient.) If after having a sample of your blood taken
to check on that substance you receive a phone call from your doctor, what is the conditional probability that you have the medical condition? (Provide a numerical answer in decimal form.) (Show and explain your work.) L5) (10 points) Toss a. fair coin :00 times. Use the central limit theorem
to ﬁnd an approximation for {he probabiiity that the number of heads is at
most '55. (Provide a numerical answer in decimal} form.) (Shaw and explain your work.) 'i
6 ‘
(5%:
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é «Law I “<1
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5’“ g‘ g 9»;
Rx? W 3' 4‘3 z .MJ—v“ f— 8 E Eable of the Siandard Nommi Distribution 919 TABLE OF THE STANDARD NORMAL QESTRIBUTION Areas under the Standard Normai Curve from ~00 :0 2: (see Figure 8.1}. 0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5754
0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
0.6 .7258 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7518 .7549
0.7 .7580 .7612 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
0.8 .7881 .7910 .7939 .7967 .7996 .8023 .8051 .8078 .8106 .8133
0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545 W
1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706 1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .991 .9913 .9916 2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
2.6 .9953 .9955 .9955 19957 .9959 .9960 .9951 .9952 .9963 .9964
2.7 .9965 .9966 .9957 .9958 .9969 .9979 .9971 .9972 .9973 .9974
2.3 .9973 .9975 .9976 .997? .9977 .9978 .9972: 599%; .9980 .9981
2.9 .9981 .9932 .9982 .9983 .9984 .9934 .9935 .9935 .9985 .9986 ...
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 Spring '07
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