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Unformatted text preview: (BLUE) §0LJW€N§ 1. (57 pts.) This problem concerns making $1 bets on the socalled “2—spot” game in
“Classic Keno” at most video lottery machines in the Four Aces Casino in Deadwood.
You, as a gambler, get to choose from 2 numbers (or “spots”) from 180; the video lottery
machine then chooses 20 winners and 60 losers from the numbers 180. What you then
win depends on the number of spots you choose and the number of matches (“catches”)
you have with the 20 winning numbers. In particular, if both of your numbers are among the 20 chosen by the video machine (so
that you “catch” 2 numbers) then your net winnings are $13. If you match (or “catch”)
just 1 or 0 of the 20 numbers, then you lose the $1 you bet for net winnings of $1. Letting W = Net Winnings in a single $1 bet the probability mass function for W can be
shown to be a. In a sequence of such $1 bets, what’s the chance yOu ﬁrst catch 2 spots (i.e. win $13) on the fourth $1 bet?
2?? 3 17 a.
(is) (an; ’ .047? b. Find yw = E (W) , the mean winnings in a single $1 bet. A:— '95))(g—17)4'($'3)ﬁ’ 2 “j i": 314 3’6 3/: —$ 0.1;: Ht‘ c. You make a few; hundred $1 bets. Your average winnings in these bets should be
roughly equal to what? continued d. Find 0'“, = ,[Var(W) = ,/V(W) , the standard deviation of winnings in a single $1 bet. 0.2. [email protected])L(;§r:l) 4— gl$)z<§‘72)] ' (*4 32—:
z H.076 5:?1 c. Find the mean for total net winnings when placing 100 consecutive $1 bets. £(U(*'"* View): é(‘dt)* "'4' Eﬁdm)
= W H 2 f. Find the standard deviation for total net winnings when placing 100 consecutive $1
bets. Vat/~(U(4.+OJND)';L¢ Var((«)()+* Vtr‘hdtaaB
: (00 (".0767 \lVét/‘(wMuk W40») 1:: V g. Again, consider placing 100 consecutive $1 bets. / l If you win 8 times (and, consequently, lose 92 times), then your total net winnings are
‘ 8($13) + 92($1) = $12 > 0. Likewise, if you win 7 times (and, consequently, lose 93
times), then your total net winnings are 7($13) + 93(—$1) = $2 < 0. Write, but don’t evaluate, an exact expression for the chance of at least breaking even in
total net winnings in the 100 bets. at N: am}; (earlorphm) / N BINDMwl n;/()O/ P: /?/?(—G
{)(Ne’l'w'mrﬁngs'io): VCN28) .
(7 W19 H r E:(/;D) (5/7} 9(/' 3/6 . 2. (12 pts.) Consider the time, in minutes, between consecutive vehicle arrivals to the Mt.
Rushmore McDonald’s driveup window. As mentioned in class, some former students of
mine determined these times — at least in the midmoming on a weekday — to be well
modeled by the density f(x) =§1e"m3, x Z 0 Suppose a customer has just arrived at the drive—up window. What is the chance the next
customer arrives between one and two minutes later? 7—; 4L“,& Jo Mxi’awt‘n ‘
7 / “ ’7‘
r 5‘ x/3 ’ 7c ’7 19([57—51): 3L %a Jan >——€ —X’3 4/; ,e
: é +& 5 @ 3. (12 pts.) The Carleton College Arboretum, where I would go for runs in the summer
while living in Northﬁeld, MN, has pesky mosquitoes that I would sometimes
accidentally swallow. If I would swallow 5 mosquitoes per hour of running on average,
estimate the chance that I swallowed 3 or more mosquitoes in 1/2 hour of running. Ml Pom V1. HWW/ A: 5/; Mt’ﬁfUIB/h‘1t' A’Q‘r M / 7 j
(péng)= 2’ @069; g ,_ 6’56. 23 5! 090
a $54; 4. (12 pts.) I purchased an 8.2 ounce bag of Reese’s Pieces Peanut Butter candies a while
back and counted 148 orange, 78 brown, and 83 yellow candies. If one were to randomly
draw out 9 pieces of candy, what is the chance of obtaining 3 or fewer orange candies?
Write, but don’t evaluate, an exact expression for this chance. K‘r‘l N;.ﬂ0mv~(j¢ My (6!
I4? {4/ 0W “Wm”?
FOUSB’): <0)(‘i + 5. (12 pts.) The fraction of successfully manufactured parts coming off an assembly line
in a work day is wellmodeled by the density function 8x7 forOSxSl O elsewhere i ' ﬂn={ a. Compute the mean fraction of successfully manufactured parts produced in a day. 5’
#:5X(Q(K)JK : S‘ KCXX7)J7 1 8x243
l)’ o a 7!
= %/D:%~°[email protected] b. Compute the standard deviation of the fraction of successfully manufactured parts
produced in a day. ...
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 Summer '04
 JOHNSON
 Statistics, Probability

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