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Unformatted text preview: UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Statistics and Applied Probability
PSTAT 120A, Probability and Statistics I
Midterm Exam (100 points), 2/10/09 YOUR NAME: ........................... YOUR PERM NUMBER: .........................
YOUR TA’s NAME: .........................
YOUR SECTION TIME: ............................ IMPORTANT: The exam is closed book. Calculators are OK, except cell phones. A formula sheet is provided at the end (you do not necessarily need all the
formulas) and space is given to write your answers. Please write all your answers
on this exam. You are given 75 minutes to answer 5 questions. Please justify your answers. A numerical answer without explanation is not
sufﬁcient for full credit.
When you turn it in, please show us your picture ID. Questions start on the next page. GOOD LUCK! ix/‘v
”306330 WW 1. [10 points] What is the approximate probability that there are exactly
350 days out of 365 in Santa Barbara when no traﬂic accident occurs, if we know the following: o The probability of having at least one traffic accident is 3 percent
every day. 0 Whether or not there are accidents on some speciﬁc day(s) has ab
solutely n0 affect on whether there are accidents on another day. You MUST use an appropriate approximation for the probability distri
bution involved. (Hint: Check carefully, if the conditions of the approxi
mation are satisﬁed, and if not, try to reformulate the problem slightly.) I: “DEE 0&6?) W] ml gawk l {Leaﬂet/in]
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1A : 363 le :: [06:3 large; vb) small in :> 056 [DO'I'SSOVU approximatimm UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Statistics and Applied Probability
PSTAT 120A, Probability and Statistics I
Midterm Exam (100 points), 2/10/09 YOUR NAME: ........................... YOUR PERM NUMBER: .........................
YOUR TA’s NAME: .........................
YOUR SECTION TIME: ............................ IMPORTANT: The exam is closed book. Calculators are OK, except cell phones. A formula sheet is provided at the end (you do not necessarily need all the
formulas) and space is given to write your answers. Please write all your answers
on this exam. You are given 75 minutes to answer 5 questions. Please justify your answers. A numerical answer without explanation is not
sufﬁcient for full credit. When you turn it in, please show us your picture ID. Questions start on the next page. G. 0 OD Ll (3 K. ! l. [10 points] What is the approximate probability that there are exactly
350 days out of 365 in Santa Barbara when no trafﬁc accident occurs, if
we know the following: o The probability of having at least one trafﬁc accident is 3 percent
every day. 0 Whether or not there are accidents on some speciﬁc day(s) has ab—
solutely no affect on whether there are accidents on another day. You MUST use an appropriate approximation for the probability distri
bution involved. (Hint: Check carefully, if the conditions of the approxi—
mation are satisﬁed, and if not, try to reformulate the problem slightly.) QUIZ “DEG+4 lCCLS+ :l. OLCCH‘JLu/xﬁj :: 303
F 2: ”Bl—“0 “CC’WSMSJ '“ l" lPEC’Kt léﬂfﬁl one amideail l l \,.C(J ”(31% W236§ ,———” l
M ’0 7/ a n p 61’ l ”L R? ﬂange»  14—13%) \2 $0516 ' IMWWW‘M 2, [10 points] Santa Barbara has population 100,000. There is exactly 1
percent of them who knows probability theory well. I call randomly 2000
dzﬁerent persons. (Assume that every person has his/ her own telephone).
What is the probability that exactly half of those I call knows probability
well? (You do not have to give the ﬁnal numerical formula. It’s ok to give
any correct formula.) . .4. oar;
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L"‘l‘®‘lool CallS ) QC} Ok) l7) 3. [30 points] (a) We roll three fair dice. What is the probability that the sum of the
numbers is 16 given that all the numbers are even? ( b) We roll three fair dice. What is the probability that all the numbers
are even given that the sum of the numbers is 16? (c) The ﬁrst two dice are still fair but there is a 20 percent probability
that the third die is not fair. If the second die is not fair then it can
only land on 1 01' 2 (with equal probabilities). We roll the dice and
observe that the sum is 3. What is now the probability that the third
die is biased? é+6+Ll .
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\“/ 3 *3 moi roll 3 llQLSQM :33 4. [30 points] There are ﬁve people in a very dark parking lot trying to
ﬁnd their 5 cars. (There are no more cars in the lot.) It is a very safe
neighborhood so nobody ever locks the car. All sit into different cars at
the same time; What is the probability that there is exactly one car which
has the right driver in it? (Hint: The inclusion—exclusion formula can be
useful.) SUP lac) 5Q, ‘ PCPSOA A 55+": M Co»? A y Tine/‘6 are, Cl WaWS ""l‘O’ CLVPQVlC’gQ) pﬂi‘gpg‘f’) l; C; D! E: 30
5; . I’ ' Had} , i“E:5‘E€,55 0% 5453+ 5“ lb .5555 5555155" (:53? So ﬂue5‘6 05%, was was 50 at E
people m S cars 80 +th
mac/“5’9 0W6 W30 5% my Wig/5,5553? 65:35“, TMFE are, g1 may; 35 can: :5) i555»: P505354»? é “Heal €XCLC%!V} (jhey InmeI/L, 55555 55 55» 5:555 5:
LIB :5: 5‘ [20 points] Pied has three good friends: Alex, Bill and Charlie. Indepen—
dently from each other, they will visit Fred sometime next week for just
one day. They all pick day at random for their visit. What is the probability that some of the three friends of Fred will meet
at Fred’s house? (3 ...
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 Spring '09
 ENGLANDER
 Statistics, Probability

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