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Unformatted text preview: (W! m) Saumw 1. (57 pts.) This problem concerns making $1 bets on the socalled “2spot” game in
“Classic Keno” at video lottery machines in Rapid City. 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 $14. If you match (or “catch”)
just 1 or O 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 $14) on the fourth $1 bet?
2?? 3< I? f
( 3/2.) “ins ‘ b. Find ,uw = E(W) , the mean winnings in a single $1 bet. ,u: (~31)(g—:>*@/+)(§%>= 4? 31— c. You make a few hundred $1 bets. Your average winnings in these bets should be
roughly equal to what? meme mm»;sz 5km;ch be W M, Ca.
x — 9.2M continued d. Find O'W = JVar(W) = ‘/V(W) , the standard deviation of winnings in a single $1 bet.
I
Z L (7 a
52: [E30 (27> * a”) “ ('31;
g (2 '71 51
(T ; (“"61 a at 2.97 c. Find the mean for total net winnings when placing 100 consecutive $1 bets. é(a)'4—~+— who) :1 50304" “"" 1 [OD (" "l. 2'1 2/6
'—~
f. Find the standard deviation for tota  u nnings when placing 100 consecutive $1 bets. Var (w + a...» "3‘" WM M— Ma»)
= lab 07—52(5) g. Again, consider placing 100 consecutive $1 bets. If you win 7 times (and, consequently, lose 93 times), then your total net winnings are
7($14) + 93($1) = $5 > 0. Likewise, if you win 6 times (and, consequently, lose 94
times), then your total net winnings are 6($14) + 94($l) = $10 < 0. Write, but don’t evaluate, an exact expression for the chance of at least breaking even in
total net winnings in the 100 bets. * lefl' A) : 19—0115 (owl—or? {603 / N (3 ($IN'W74’ “>U’D/ V: M/Za 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) =ge’4x/3, x 2 0 Suppose a customer has just arrived at the driveup window. What is the chance the next
customer arrives between two and three minutes later? 7'; ﬁne, 41> rex't—arviwl
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ﬂzeresﬁ }’ get/3.4... e_ x” '2’ 70,3 x92. 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 4 mosquitoes per hour of running on average,
estimate the chance that I swallowed 3 or more mosquitoes in 1/2 hour of running. W Pow V7. Haw» : A —. 7. wwﬁo—J Hawk.”
N = 1L MS fun/172:4 smllMoe
M J 7. L . '11»
Fax/23): 2%e‘=l~_ﬁye “~@
6:3  jao o K5. 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 7 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. 5. (12 pts.) The fraction of successfully manufactured parts coming off an assembly line
in a work day is wellmodeled by the density function 9):8 forOSxSl O elsewhere a. Compute the mean fraction of successfully manufactured parts produced in a day. M: ‘3 j‘: x(‘?7g8) 47‘ 3 S; 773Jx n ‘7'K/o/l— 1—0 3 0 b A) /o b. Compute the standard deviation of the fraction of successfully manufactured parts
produced in a day. 7.
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b
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3 l7 12>ch __ 1
{Sb x i} (a) ...
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 Fall '04
 JOHNSON
 Statistics, Probability

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