STAT 315
Spring 2026
Midterm I, version A
NAME:
fiLl^i,an frtslu,,
STUDENT #:
3so7qq (s
INSTRUCTOR & TIME:
€qt
Instructions:
1. You have 90 minutes (1 hour 30 minutes) once you begin.
2. You are a.llowed one 8-5 x ll-inch, double-sided formula sheet of your own creation. No other outside
materials a.re allowed.
3. For firll credit, show all your work and provide justification for every steP.
4. Write your solutions clearly and legibly in the area following each question. OnIy write on one side of
the paper. If you need more room, please staple an additional piece of printer paper (available from the
proctors) aod clearly indicate the problem you a,re solving on each extra page
5. You may leave numerical aJulwefii a.s unsimplified expressions (for exa.rripie, sums of fractions, factorials,
or binomial coefficients such as Cp or (:); equivalent forms (ocpanded or simplified) will be accepted.
6. Retsrn this er<am sheet when ;rcu are done or at the end of the exm. Show the proctor your studeat ID
and sig. yor11 n&ne to the roster to ensure your exam has been collected. Only the exa.rn sheet vrili be
gaded.
7. The exam has 100 points across five problems.
1. (20 points) Alr experiment consists of rolling a four-sided die and then flipping a mir. The four-sided die
has numbers 1,1,2, and 3 on the four fares; the number "1" appears on two faces. The coin has numbers 1
and 2 on the two sides.
(a) (4 points) List all the sarxrple points of the experiment.
T_
tl','3 ,[,,rJ {z t3 {r,4 , {3,,j, ft Z
jl
{ --66
(b) (4 points) Let ,4 be the evetrt uthe sum of the two numbers facing up is even". List all outcomes
belonging to .4.
h={(t,t), (z,z),bDJ
I
A
I
il ?
(c) ( points) Assuming that the roll and flip are independent, every face on the four-sided die has the
same probability of landing face up, and the two faces of the coin have the same probability of landing
face up,6-nd the probability ofA using parts (a) and (b). Show all your work.
PG)=
l
AI
J
lrl = 6
Z
(d) (4 points) Let B be the event "the four-sided die shows a.n odd number". Find the probability of A
given that B occurred. Show all your work.
A
,
(r,)9
nD--[C,,)
b-- ft ti*<,cld' d,e sl^r* od)'r
n = ft ,r),(t,r) r (3,) t (3tr)3 , td= 1
0
o(o a)=
P( a no)
Z
P(D)
1
PtB)-2-
3
€
X
i'3
(
Z
(e) (4 points) Are A and B independent? Justfi your answer.
1r" h a^) 13 l.
P(r ir:) = c(
h i^/ep^d'nf,
P6^s)'PG)t{rs)
t)
P(to)= ). , P(o) =
I
[nen66 + A^
',
) b^<
i"c/rg^/*t
f$ nrs)'= lann)
3
trb)t(B) -- +"+ --
i
tsl
I
{.
-l: \,
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3
2' (28 points) A box contains five red balls and seven white bFlls. 11 this. problem, you may use properties
of probatrility distributions from the textbook without proof.
(a) ( points) A random ball is drawn from the box, its color is recorded, and the ball is placed back
in the box' This is repeated six times, where each draw is independent of all others. Let X be the
number of times a red ball is drawn. Wlat axe the na.rne aud parameter vatue(s) of the distribution
of X? Provide justiffcation for full credit.
\: fr,v) bql19 d't,va ->drst11^n^ \(, = L,^u*iol A'<h^tb,Ltu^
f)vo-,ol e,r+ (n= 6
t_
)
tP - z )
(b) ( points) What is the probability that X is three or more? Show a,ll your work for fulI credit.
?(*>,t = l- P(, Z )
F(X
t) - p(x o
,-lrefel lilt+t'(d @E)?Gl I
J
?( \)-
(c) (4 points) A random ball is drawn ftom the box, its color is recorded, and the ball is placed back
in the box. Each draw is independent of all others. This is repeated until three red balls have been
drawn. Let Y be.the mrmber of total drax,s in this process, i.e., the draw on which the third red ball is
drawn. Wtrat are the na.me and parameter value(s) of the distribution of Y? Provide justiffcation
for full credit.
,d,$Fnb"kl' ]3ea -. '|Mlati'a b,n,-'o:l
a -
-,
\
)/r,
,f:
)r
(:
rrpdqfi(
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Diuc,,"'al
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)
/
\
(d) ( points) If the third red ball has not been drawn by the fourth draw, what is the probability that
the third red ball is drawn on or before the tenth draw? Show all your work for firll credit.
rr(rs,l) = ,D6,1,) + 'D0 'q)'*s{r=u) t **=')t'4--')
+ nryl--s)+
,(+,
)Gl ,(ilG )
@c
1
(,
'0&fr(3)e;
,Q)?*
(e) ( points) A group of six balls is drawn at ra.ndom from the box without rcplacement. Let Z be the
mrmber of red bals in this group. What are the na;ne and parameter value(s) of the distribution of
Z? Provide justiffcation for firll credit.
:
6l tst ^ lJ i2
llfv
yo,aefat
^.r^,,(
r\
V1p.,1towhrU= t7,
^--
G, K = 5')
(f) (a points) What is E(32 - 2)? Show all your work for full credit.
v\r11 t;^.a,ij of .r1r'fufiu )
ur{z), ^C!")
"c)-,
.
W
t5
2
E (tZ* ,)= lf(.)
a)
Z
:;hq
L
(g) (4 points) What is V (32 - 2)? Show all your work for full credit.
v(>z-z) = lv(r)
v(z):
v(z
C;{ f) N-t
): 9_
Z
G)€ft)
I
q
1 2
G)(+\)
3. (18 points) Thtee ride-shari.g mmpanies Rl, R2, and R3 operate in a city. A randorrly chosen ride is
provided by company Rl with probability Q[" by R2 with probability 0.4, and by R3 with probability;[!
The probability that a ride is cancelled given that it is provided by Rl is 0-2, by R2 is 0:&, ana bEry i.
0.1.
(a) (6 points) Find thc overall probability that a ride is cancelled. Show ail your work for fr.rll credit.
?(c)= 0G .R)PG,), p c lit2 P(P, ) + ((ctiz
ftt)-- (, ,G
t
?(p: )
b z)(o-r){.0-r)[o.r)
f[ rt,) = r. 5 ?(ctrz). o L
([e,\-- o'Y 0 (un")'- o z5
P[rz)-- u-1 P(ct Rt)-- o'l
(b) (6 points) Find the probability that a ride is provided by Rl OR the ride is cancelled. Show all your
work for fi:]l credit.
P(o, Il c )
tr&
VL
frcr re,) e(tz,\
e(r, , c)
r(2,) r FG) - Q('2, n'l
o -5+ (o.z\,-s)r (,'z)(,-,)r 6o-) - ("'.b)P 5)
(c) (6 points) Find the probability that a ride is provided by Rl given that it was cancelled. Show all
your work for full credit.
P('z-,|c)=
Q(P,nc\
ilc)
p(LtP,)r(2,)
)
t= I
I
I
@.ro) (" ,)
Q..)[. s)r 1".2s)[0.") +(0. l)z
f(qv) ?0t,)
g(2, tc\
4. (16 points) On each turn of a ga.me, two fair six-sided dice are rolled independently. The player's reward
is 0 if the sum of the two dice is 9 or less, and (sum 9) if the sum is 10 or more. l,et X be the reward on
a single turn of the ga.me. X ...-< t1
(a) (4 points) Find the pmbability distribution of X. Show all your work for fi:ll uedit.
X-- tt^o'l^ o- o<r1le'lw'n
, lol"\ suw: = [ 'Lr3 1u,Sr{r7,9r1, lo/tt/ tz 1
t
n,/
)/76
Zhe'l3L
Y- {0, ttz,}t
2(x, o) = (s,* < r")
|
- p(su,^)Lo)
tr\ - --2-
^,/
t-Cllu', )',,-+n l'g''asu
+
?(x'-z\-- (( s"^-n)-- *
P(r=r) = ffs.,*:ru):
P(t'=z) '- '/rs
f\.t\ = 'lz(
g(r, l) -- F(su^= r)-- +
(b) ( points) Compute the orpected value E(X). Show all your work for fuIl credit.
e(x)=t.lf')
o
t
6
J t 2tz
I
3r )6
'lr) tZ
'l,s
) , t('lro)
)
(c) ( points) Compute the variance V(X). Show all your work for full credit.
v0 ), t ( Y z\)-
v(+ (
_)- t o1
+
9 )1"
L
I
r(
Z
)-('.rft+
3
Z
t6 )
ry
T
(d) (4 points) Find E[(X - 1)a]. Show all your work for
/-'
E
x-l ) \
U
ry
t\
LG\) - L(o)
5. (18 poinLs) It is dismvered that four items in a batch of lrwelve are faulty. The items are ordered randomly and then tested one at a time. In this problem, you may not use known properties of probability
distributions; that is, you must derive the answers using laws of probability.
(a) (6 points) What is the probability that exactly three of the first five items tested are faulty? Provide
justification for each step for full credit.
)
lz ittv,f ,< +tst9 ) b+ql vtals l- 'an/on!
/ tZ\
t7'lt)
char's< 5 rk'^s h' lrsf 'uf of
fo
\ {r,'t, ffo', * Mrl ft^,ttJ9->l')til
"?t
"k*
(\)
r (uose ) P^, l\s
^wrt ''t '4 =
3)
oflq, l*u kr+S 4{p nslrlLal ,"u lAus. ff * * fw^ lL' ofl*v ,9
l\e
^r*
fl-ot a<
&"1/1-> fula(.way h, ch,,,se, Z,nuq-h'lD '[e-r fwq f ' e)
7) fr,* t1c 5 *e lectlwe Tl
"rt
P(e,-,+t1
) {,"ty)= [t) (g )
IZ
C.
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)
(b) (6 points) Given that exactly three of the first five iterns tested are faulty, what is the probability that
the remaining faulty item is found on the sixth, seventh, or eighth test? Show all your work for full
credit.
.
k- 3 &"d ,^ h,sl 5
?(x*a la)
- r(x -6)
;)G)f+l
+
r6 rI u
q
3
2(x-t lt\ = l(x=t\ ? (t \ln )ti )
f
t(r q lr)= f(r 3
)-f) ftI
L
I
Y\iara 0
7
+ rL< 0 wlzLitihes
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(c) (6 points) What is the probability that the last (fourth) faulty item is discovered precisely on the 10th
test? Show all your work for firll credit.
)
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(1. ofl,'tt '(L*e
q
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