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Unformatted text preview: awwﬁZA j on) Circle your ﬁnal answers to each Problem. Problem 1 (6 points) a) Disease Y is a disease caused by a sex—linked recessive mutation. A son of a carrier of
disease Y can get disease Y from the carrier. Some disease Y cases, however, are due to
spontaneous mutations in sons of mothers who are not carriers. Jessica has one son, who has
disease Y. In the absence of other information, the probability is 0.4 that the son is the victim of a
spontaneous mutation and 0.6 that Jessica is a carrier. There is a screening test called the CK test
that is positive with probabilityif a woman is a carrier and with probability; 0.3 if she is not.
Jessica’s CK test is positive. Find the conditional probability that Jessica is a carrier given that
her CK test is positive. Round your ﬁnal answer to four decimal places if necessary. (3 points) ' ' P(rlckon MTV)" 0:3
(“m W P(T‘Ic‘)" 0.7 ~ Paar): ,7 FCCIT): ‘ 0.9% __,_ 0.3 ‘8‘88
0.9» +0.11 b) Suppose that A and B are two independent events with P(A) = 0.2 and P(B) = 0.5.
What is P(A or B)? (3 points) WA m 6): WM Maya/4M 8) WW 8): WW5)
 ' : [0,»)(05) : do" Problem 2 (6 points) a) Each of us has an ABO blood type, which describes whether two characteristics called A and B
are present. Every human being has two blood type alleles (gene forms), one inherited from our
mother and one from our father. Each of these alleles can be A, B, or 0. Which two we inherit determines our blood type. Here is a table that shows what our blood type is for each
_ combination of two alleles: Alleles inherited Blood ty 6 We inherit each of a parent’s two alleles with probability 0.5. We inherit independently from our
mother and father. Elise has alleles A and B. John has alleles A and B. They have two children.
What is the probability that both children have the same blood type? Round your ﬁnal answer to
four decimal places if necessary. (3 points) b) A full deck of cards contains 13 diamonds among its 52 cards. Among the 10 visible cards
displayed on a table, 3 are diamonds. What is the probability that the next two cards drawn (from
the 42 unseen cards) are diamonds? Round your ﬁnal answer to four decimal places if necessary.
(3 points) ' i6“ PM W 0): 1969+ D)“P(2hd V) 5+ p).
s (a) (a) : D.O§zzwsaw Problem 3 (6 points) Consider the following information about random variables X and Y: ux=10 6X=6
HY=11 GY=9
pxy= 05 Consider another random variable: R = 0.8X + 021’. a) What is “R? (3 points)
ﬁx : [050M Jr [0.2) M
.: (0.8) ﬂo) +/o.1)[n) = b) What is 0R? Round yOur ﬁnal answer to four decimal places. (3 points) (2: /0,s’)>(4 /0‘L)Y
12: F 4’ C
6:: 6,2: 4’ ‘2' 4’ 163461439
2 (0,5395ng MAE; + 2fo.r)/o.‘6)6gzo®6v =/o,<69’/5+m«>5 z/ww/oﬁsxa/oaxo : 17w
épW Problem 4 (6 points) a) The proportion of the American adult population that supports candidate Hayworth is
p=0.26. A SRS of 15 adults asks if they agree with the statement “I support candidate
Hayworth.” What is the probability that exactly 4 of those surveyed would agree with
that statement? Round your ﬁnal answer to four decimal places if necessary. (3 points) lg) _ ls‘. Maw/Emu}? Lt " tun! — 7;
Waxzxa x)?“ ’ Bé
Past 9‘ 47%th I! 615M) : /D.26)H(o.7t+)“ WW L’ W): ‘35 [old/0.76:] == annsmos ' it: 0.17.73 ' b) The insurance company sees that in the entire population of homeowners, the mean
loss from ﬁre is p_= $700 and the standard deviation of the loss is o = $400. What are the
exact mean and standard deviation of the average loss for 14 policies? (Losses on
separate policies are independent.) Round your ﬁnal answers to two decimal places if
necessary. (3 points) Z: 7'; A+b°4...*N] = 67% 4638+. . Z’ I :: J/IZ 7’ {L} 7.
(ii/AAA/GQQW] 6%) 6A JréugBibvl’n
(L L(52+ 671,. '—
y 25/4)MJ<’.JM+
saw/m
: e M2 :JTDM/t 7L MB +v. Problem 5 (6 points)
Round all of your ﬁnal answer to four decimal places if necessary. The joint probability distribution of X and Y appears below: a) What is E(X]Y=O)? ﬂWeP%%wzsww?i  r
___. 1,
a...» '
l b) What is E(YX=4)?  anw:[email protected]%+%%0 c) What is E[E(YIX)]? £6“) a E [5&0] ' 9w) : o (0.9+ 2&5) LL 29% a) The proportion of the American adult population that supports President Obama is p=0.54. A
SRS of 7 adults asks if they agree with the statement “I support President Obama.” What is the
probability that a majority of those surveyed would agree with that statement? Round your ﬁnal
answer to four decimal places if necessary (3 points) ( [ggqlﬁtoﬁéi EoS‘Og/orrei} (Z)@»WS{O.%{I* [2 Sufﬁx/J] a w
. g; 1 Problem 6 (6 points) 4.39.:— // +1 // : QFXQ‘M'LWN : 0,576? b) The proportion of the American population that has disease Z is p=0.02. If 60 people are
randomly selected from the population, what is the probability that at least I of them has disease
Z? Round your ﬁnal answer to four decimal places if necessary. (3 points) a A
Watt MW it) = (0%)“)
5; AL..
VCMW My?) :2“ [—r /OFI%)6O Problem 7 (6 points) a) The proportion of the American adult population that supports candidate Paul is
p=0.21. A SRS of 9 adults asks if they agree with the statement “I support candidate
Paul.” What is the probability that at least 2 of those surveyed would agree with that
statement? Round your ﬁnal answer to four decimal places if necessary. (3 points) X: quxvmm [WM 107%): Waistcoats award Wu“. :— (0~21)(0.74)8 + [0.1) )(0,74)g+‘ , t: d zomzmﬂ I b) Suppose that A and B are two events with P(A) = 0.5, P(B) = 0.4, and P(BIA)=O.2.
What is P(A or B)? (3 points) W 6» 9): P(ANW’E) ’Pému 9) pm a): maze)»
: 0M / 0.) : [06)(0'23 1 1:0, Problem 8 (6 points) a) The voters in a large city are 50% white, 30% black, and 20% Hispanic. Candidate Smith wins
40% of the white vote, 70% of the black vote, and 20% of the Hispanic vote. What is the
conditional probability that a voter is black given that the voter voted for candidate Smith? Round your ﬁnal answer to four decimal laces if necessary. (3 points) 0,1. Fag/ﬁt); “93;” : 0.954%?» :: L
[b.HOdH (Log V b) In a large city, 60% of the voters are Democratic, 30% of the voters are Republican, and 10%
of the voters are Independent. Candidate Jones wins 60% of the Democratic vote, 30% of the
Republican vote, and 40% of the Independent vote. What is the probability that a voter is a
Democratic voter or a voter who voted for candidate Jones? Round your ﬁnal answer to Log decimal places if necessary. (3 points) a
06 06M W a”
’95—‘36? 0'3 0’3“ \ 0. 0,7,) WW
%W 0.02 my 019‘: Mpg/n may art/Madge ya): man wt) 45mm”) 3 0.4.» + mm a 0.3; 10 Problem 9 (6 points) Round all of your ﬁnal answers to four decimal glaces if necessary. The probability distribution of random variable X is given below: Value of X X 4
Probability a)Whatisa;? ' ﬂy: 0[o‘L)+7/(O:234W60.6) == 21% 53;: (07.2.3)1(Q1)+(1'2;%)L(0n1) +(H4'2x‘831/oé) .’ '
’ b) Consider another random variable: Y = 0.5X— 0.2. What is pry? I
My : mam  0.2.
'5 /O,§) (7/73) “0'1” .3 c) What is 0'; ? 11 Problem 10 (6 points)
Round all of your ﬁnal answers to four decimal places if necessary. In the language of government statistics, you are “in the labor force” if you are available
for work and either working or actively seeking work. The unemployment rate is the
proportion of the labor force (not of the entire population) who are unemployed. Here
are data ﬁom a government survey for the civilian population 25 years of age and over: Total Po ulation In labor force Employed
Not ﬁnish HS 20,000 13,000 12,000 HS degree, no college 42,000 38,000 36,000 Some college, no 37,000 34,000 32,000 college degree College degree 42,000 40,000 39,000
12,5000 Moon a) You know that a person is a college graduate. What is the conditional probability that
he or she is in the labor force? __
0, Gib zsroagw m? 6%) 7 “0’0"” "5 W W’OOO OHSZW b) You know that a person is employed. What is the conditional probability that he or
she is a college graduate?
0. @773qu ﬂaw ‘5 c) You know that a person is in the labor force. What is the conditional probability that
he or she did not ﬁnish high school? ( : . ‘5 0.] 0")
{253000 P M ) Iéjéoo Gaﬁm 03> 12 A college decides to admit 1800 students. Past experience shows that 80% of the students
admitted will accept. Assuming that students make their decisions independently, the number
who accept has the B(1800, 0.8) distribution. Problem 11 (6 points) a) What are the mean and standard deviation of the numberX of students who accept? Round
your answers to four decimal places if necessary. (3 points) I
‘ :7. DD {03): ‘ ruN h? W96?) n? [’3 > X ( ) > W064) z’l$033(0$3(03, :‘ /;ﬂ7o54173 b) What is the probability that more than 1450 accept? Use your ﬁnal answers ﬁom part a. Use
an actual value from Table A. Round your ﬁnal answer to four decimal places. (3 points) Problem 12 (6 points) 13 A SRS of 700 adults asks if they agree with the statement “I support President Obama.” Suppose
that 60% of all US adults would agree if asked this question so the population parameter= 0.6. a) What are the mean and standard deviation of the sample proportion 13 ? Round your answers to four decimal places if necessary. (3 points) [O'EMOWD' '700 "=— 0,0/ﬁ5lé am, b) What is the probability that the survey result differs from the truth about the population by '
more than 2 percentage points? Use your ﬁnal answers from part a. Use actual values from Table A. Round your ﬁnal answer to four decimal places. (3 points) A P  < 0101
DID/0.42
0,01‘6‘3 0,58 ( US$09 . A'ﬂA
I ‘73 f
0,0l‘6S < 4 «Ltﬂlﬁ 4‘2 4 1.0%”)? 4.075 4C 2 < M3
(9; 0.25m; 59 a 05W 14 Each of the 14 multiple choice questions is worth 2 points (28 points
total). Circle your answers. ' 1. You are interested in political attitudes among the 3000 members of a
sorority. You choose'60 members at random to interview. One question is
“Do you support candidate‘Y?” Suppose that in fact 12% of the 3000 y“; :w members would say “Yes.” Here, can you safely use the normal F: Din,
approximation for the sample proportion when an SRS of 60 individuals
A) $21511 the populatlon of 3000 1s drawn? h? 3 m _ h (V?) Z W NO ‘ é0(0,\1)= 7;;
'  M 2. Deal 40 cards from a shufﬂed deck (52 cards) and count the number X of red cards. The countX have a binomial distribution. Toss a
balanced coin 20 times and count the number Y of heads. The count Y at»? have a binomial distribution.
A) does, does
' B does, does not
@ does not, does D) does not, does not The table below shows the political afﬁliation of 1000 randomly selected
American voters and their positions on the school of choice program. ‘ Political Party . Position Democrat Re ublican Other ; Favor ' 260 E 120 3 240 r _ Oppose 40  240 100 ‘1
2w 3. What is the probability that a randomly selected Republican favors the
school of choice program (rounded to two decimal places)?
A) P(FR)=0.12 ' Mama " Wm;— D) P(R  F) = 0.36 15 4. The weight of medium—size tomatoes selected at random from a bin at the
local supermarket is a random variable with mean it = 10 oz. and standard
deviation 0': 1 oz. The weights of the selected tomatoes are independent.
Suppose we pick four tomatoes from the bin at random. and put them in a bag. Deﬁne the random variable Y = the weight of the bag containing the
four tomatoes. What is the standard deviation of the random variable Y (rounded to one decimal place)? \(7 Amie—H) 6 __ W ,DL
= . 1. ~“’ “ a ’
A) 0y 0.5 oz. 2,6736 Jr
_ 6v v" A B 
B) O‘y— 1.0 oz. 1 1
: 6‘ + 6 4w _
@ 0y=2.0oz. : *(\)t ;H D) 0y = 4.0 oz.
E) A number other than any. of the other choices listed. W ; 54g 5. You are interested in political attitudes among the 895 members of a v) ~91)
sorority. You choose 40 members at random to interview. One question is
“Do you support candidate Z?” Suppose that in fact 40% of the 895 members would say “Yes.” Here, can you safely use the Bf 40, 0.4)
distribution for the count X in your sample who say “Yes”? ‘Yes W 20 Me > h B) No .  . A ,
. What CLD)C‘+O) :—: 800 v
6. LetX equal the count of the number of heads in four tosses of a coin. Here,
A) is a continuous random variable is a discrete random variable
 is not a random variable. (3 C) 7. The of any Collection of events is the event that at least one of the
collection occurs. Disjoint events be independent A) union, can @ union, cannot C) intersection, can
D) intersection, cannot . If X and Y have correlation p, and p _> 0, then (7)2“), > a; + 03.
A) = ~ 2 3:1'l4/l4l 9. A)
B)
0%
)
F)
10. A)
@>
c)
D) 11. A)
B)
C)
D) C? E) 16 X and Y are two independent random variables. Consider the following
statements: (1) The mean of X + Y is the sum of their means.
(2) The variance of X + Y is the sum of their variances.
(3) The correlation betweenX and Y is zero. The variance of the difference X — Y is the difference of their variances. (5) The mean of the difference X — Y is the difference of their means. Which of the following statements is correct?
Statement (1) is false. ' Statement (2) is false. Statement (3) is false. Statement (4) is false. Statement (5) is false. Statements (1), (2), (3), (4), and (5) are all true. 1 'L 1
{X4 3 64415" If X is a random variable and a and b are ﬁxed numbers (61$ 0, 19¢ 0), then 2 _
(7an 2
bO'X
2 2
[9 O'X
0;
2
a+b0‘X What is the probability that you get heads on the third toss of a coin given
that you got heads on the ﬁrst two tosses? In other words, What is P(heads
on third toss ] heads on ﬁrst two tosses)? \ 0.015625 M. 0.0625  0. 1 25 0.25 0.5 A probability other than any of the other choices listed. 12. A)
B)
C) C? 17 In a large city, 80% of the residents support President Obama, 70% support Secretary
Clinton, and 65% support both President Obama and Secretary Clinton. What is the
probability that a randomly selected resident supports President Obama, given that the
resident supports Secretary Clinton (rounded to two decimal places)? 8.45;: Pmﬁc) Paar/0mm; = Pam Mr 600)
0:81 0.7 X :: 0.63
0.93 X; 9'...“ :o,4zs!7 A probability other than any of the other choices listed. 0. 7 The probability density of a continuous random variable X is given in the
ﬁgure below: A)
B)
C) F) Based on this density, what is the probability thatX is between 0.5 and 1?
0.0625 ' 0.125 0.25 0.5 0.75 V A probability other than any of the other choices listed. Call a household prosperous if its income exceeds $150,000. Call the
household educated if the householder completed college. Select an
American household at random, and let A be the event that the selected
household is prosperous and B the event that it is educated. According to a
government survey, P(A) = 0.29 and P(B) = 0.44, and the probability that a
household is both prosperous and educated is P(A and B) = 0.21. What is the P(Ac and BC)? 8 B ‘— 0.1232 0 27 A ' ' 0'14
. C . . 0.4032 A m 0.48 W 0.79 0' A probability other than anyvof the other choices listed. ...
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 Spring '08
 ABDUS,S.
 Probability theory, President Obama, decimal places

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