$oLUnoNS
page 1o16
Math 35L, Probability
Exam 1
October 4,2OL2
Directions. Please write your narne at the top of the page. Your answers should be in simplest form
possible without a calculator. You may leave answers as fractions, differences of fractions, factorials,
choose, etc. You are encouraged only to simpiify when obvious and necessary. Please BOX YOUR
ANSWERS. You must show work for credit. There are 100 points on the exam, so pace yourself
accordingly.
L.
(14 poi,nts)
(u)
Fo. (rtr a
(S points) State the binomial theorem.
(r*1)^
^
fn\crr
(b) (6 points)
=
-*(?.)
*tJn-n
Use the binomial theorem to prove that
:'
2h(;)
let x3.5 =g
Then [x*.1)a.o I
+
V
fi7ro
.3Uf tll,
I -- 2,(?).s't.sn-"
P7o
,.t,,
Page 2 of 6
2.
(10 poi,nts)Three 6-sided dice are rolled until either a triple occurs (3 of the same value), or the
numbers 1,2,3 occur at the same time, What is the probability that the game finishes with a triple
occurring? Note that 6'6-. 6 :21,6.
;';;-" =i iiti*'W *^
T
te*-
rs n
*tr.eal
f, " E
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4
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(=2!
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neit|nt{,
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ro[^'r Sve,Lb') (fi-'@}' wificlviplc- ffir.ffi,"e-lt.O
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FrL
-
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{gf
H
=!a4"rcl-,
t(flg))"i+B
Mason Southside Dining HaiL Two of the countries represented are Spain and
probability that the students from these countries are seated together?
Italy. What is th{
,
tl^}/w e& g I ubqs h P*
st at\sr*ul lt^e" y,blr,
(VyT
tit l Iry ""ttr€ I
fi,'.,aolL""
r?tanue
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\!l:_
coisidorruyy
lolote aA iW1r*-*\ /o^^ ca,n, uot ll'cunAfiLt \
inaSl n*it
t4^r^^^ c.,n unc "unit'l h ^ uni!- t -2, t{"ll..
.,i0
S CIob wfi{+,w ll.z Pn(f
o,.d,Z wo,1 h octuj*
= |gb
I f
gprp
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Page 3 of 6
4.
(10 points)Three students meet up at Jane's Juicy Juice. Without paying attention to each other,
they each randomly order one of 20 variations of juice. What is the probability that two^of them
ordered the sarne drink?
prIU
*
WVVLflNtre*WhOaN/,Ldv-6(ltrtl"'zsara"td,n'tk,"1rrr, a" fi fl^t W,',rlL'ol i^one fu)(. A*lpred flu Sr,,rte d,rt'W,
'Tuq P(a;= t-P(E').
la)r' dl
Wa ca-i,)u^ldte ?(t ) - g2 , n..lA { vt.^,4loc'.$ vdil '48h"4
20'
€
ru,,,hq'4r*;;
l'inks
lnprs 4, a,n*s
-it,rta P(.f-)=l- tf,lt
'2t
5.
(10 poi'nts)Urn
A contains 19$red balls and 2 white bails. Urn B contains 1 red ball, and gg white
balls. A computer is programmed to pick Urn A with probability 1/3 and Urn B with probability
213. The computer selects an urn, and then a ball is drawn. If the ball drawn is white, what is
the probability that it came from Urn B?
[-r,r At
$z
u,snl urn A
r\
i5 chosc"-
B
c)1#4''
R 3 7.uuut ,2rl ba1 ls
is (jtos<*t
Yf , e^,L'^h ujnu'fg b'^tJ
$e *od P(stW).
?tslr^t). PlwF)wh
?Tw1
'
P(t w\s)l:e)
P(Ns)"f('wA)
3@
etw ls)P tb) + vFl
.9Q.?g _4
.im;
.o\.r/3
*)ttr)
.bb
= .la+ .oo33""'
*. 115
Page 4 of 6
6.
(16 poi,nts)Four professors, one of whom is Dr. Goldin, are playing poker. They are each dealt 5
cards from one normal 52-card deck of cards.
(a) What is the probability that Dr. Goldin has a full-house? A full-house is 3 of a kind, and
2
of another kind.
ll*t .&,QobL ,n"Ia hn,E. v,!etweh(a,+
rc) hc.,ri,sha^da
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(b) Dr. Goldin gets 3 of a kind, but her other two cards do
Aatch the other cards nor each
other. She discards two cards to try to get a full house"#,
or four of a kind, and receives 2
new cards from the pile. (The pile does not contain anv discarded cards in it). What is the
probability that she succeeds in getting a full house or four of a kind?
Were +Lre ke,Ait'rYl
whrl '*nb D,.' 6o//n'n
Fl4vt 'ufaur 1
olt*l
canh l4Alx /'c'*
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Page 5 of 6
7.
(20 poi,nts)Ten people are dealt 5 cards each from a standard deck of 52,
with the remaining cards
set aside.
(a) What is the probability that the first person
holds).
set of card a person holds).
I
t
El\/Sg\ €
has exactly
t
heart? (Remember a'hand' is the
rrfirsgyl
B l^eat* , at ';.iru'ch'TEnot* !- . Tle * ot[,". @u,ds 4r1Je)yg"r.era yry
u4tn fa*
l-i"e 3f
3E nou,.hea t
h t^z,e run,l
,noq.he4fr, *re ha.1- 4&
i /f, { /
Wn #fiw'
fuL
P P"i*',,o*tilwrc
(b) What is the probability that the
has exactly one heart]
second hand has exactly
t
l^e4/Lf
heart, given that the first percon
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Wpt-els Yib'le.d b /ctztssa
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f'flS\
(c) What is the probabilifii iflat' the third hand has exactly
1
heart, given that the first two
people have exactly one heart each?
(\')(?)
E
t
is probability that the last pers-on has two hearts, given the first 9 people have exactly
'ru-lil
- ff t ?
[t /
(t1- foPle
.l'vwrfteflg,4ry{.
w
7 4,renu.Nrvq *W4 l,rg=19
@r7^ WeL*e(
tufrcv pt*-
d4itYbufed
(e) What is the probability that the first nine peopie have one heart each, but the last person
has exactly two hearts? Justify your answer. You are encouraged to use your answers to the
previous
parts.
-rO\
f"* iit = f*y #ff*vv* uu't\ {
A*A
|,,a^I-
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at
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Page 6 of 6
8.
(10 poi,nts)Consider an urn containing four balls, numbered 110, 101, 011 and 000, from which
one ball is drawn at random. For k :1,2,3. Let Apbe the event of drawing a baii with 0 in
position k. Show the events are pairwise independent, but not mutually independent.
r{r = Ton,oooJ
Az= f torJooo/
?(Ai)'i
i=l'2'3
ha- teno,so3 ?h)
I
,l0tt,tor,llo/oooJ
A,AL=
ArAs= A,t3 =lo,o,o\
.
fte; Al )-w;fi,\ it
*
fun RAi)ttU-- L',L. : p(A,-Al) F,r?
i *fe pcrfvwrte
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+{^q"{- tt^t
[*uvc.r.qr7
h ,\rAB - 7o,o,oI + P(e,A, As )={
l+nue-w
( (A,) prAr) p/Ae
So A,lAarAf
a.t\e ,t*
,v.,.ptcr
.
)= * _+.+- =d * )/.t,[n,n"{
M^rh4*
rud6pen/a.*i
.