Solutions to Exam 2, White Version, Ma381, F05

# Solutions to Exam 2, White Version, Ma381, F05 - (W m Saumw...

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Unformatted text preview: (W! m) Saumw 1. (57 pts.) This problem concerns making \$1 bets on the so-called “2-spot” game in “Classic Keno” at video lottery machines in Rapid City. You, as a gambler, get to choose from 2 numbers (or “spots”) from 1-80; the video lottery machine then chooses 20 winners and 60 losers from the numbers 1-80. 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, C-a. 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 drive-up window. As mentioned in class, some former students of mine determined these times — at least in the mid-moming on a weekday - to be well- modeled by the density f(x) =g-e’4x/3, x 2 0 Suppose a customer has just arrived at the drive-up window. What is the chance the next customer arrives between two and three minutes later? 7'; ﬁne, 41> rex't—arviwl ,4 / 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 “[email protected] 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 well-modeled 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. :1 (3‘ Kl?7(v47< / b 'L 3 l7 12>ch __ 1 {Sb x i} (a) ...
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