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Unformatted text preview: ‘ Chapter 5 — Expected Value Example 14: Sum of Balls l75 ; Let X be the sum of the numbers on i balls drawn at random with replacement from » ,1; ;
this box: " H ®®® a) What is the smallest X can be? 1 ,
E>vh7€b = ‘8‘753 , _ i:'
b) What is the largest X can be? 1 50.ng: 52.50
c) Most likely, X will be around 30 6: Z . 5 I give or take i [2, c1 7 or SO_
ﬂm‘admm: 5'Moner°\\ocht;¢ ‘1 535: l7. '5 Jone Jr4w : 5 . 6—Oﬂe rollo‘qulf: : a l l' : ‘
Méum f [76’ Marie drew.) 7” [—75 ' '7‘ 5 : 65am 1 “75 'CTZnealraw : '75' 653% A“: When there are only two different values on the tickets in your box model, there is a
shortcut formula that you can use to calculate the standard deviation. If your box model is as follows, [ca/:9 ME] then the standard deviation of the box is ab 021*” (a+b)2' wave mama ﬂew H2 Em us may mo ' "f;
mama): to/ Fgquceme/(f from q box \ :theﬂ Eff—En] :ﬂbo‘  N005 w: say : 5?,an
I f1 , 68 _
(M5,: aolfu stink/175 “pad—0‘" Ha W/O re .0 lqc’en’lé’fl 15.) Chapter 5  Expected Value Example 15: Roulette Wheel, revisited A roulette table has 38 slots — 18 red, 18 black, and 2 green. In Example 8 we bet $1 on red so that if it comes up red we win $1. Otherwise, we lose $1. Assuming we play
40 times in a row, answer the following questions. Le—tx be {he Viel: 4M q~F£e( Jamar Orwe
a) What is the box model for this problem? ﬁrg‘taﬁm ('2: H0 draws w/fe‘C)lQCvaﬂ.é
[/8 ZOIE] éoIzsom b) Calculate the SD ofthis box. Calculé‘é': Mbax.
Cah‘ula'lit “(‘0‘  WW.— ’ '*
’abox :W : 'ég‘  ~$o.05 => EEKl’ “0 F3405 3% 39 . _ )
01.x = Im— ﬁlm/0’33. = $1 => SDEX1= c) There is abouta68% chance thatXwill fall in what range? bellshaped 7. m' $1 (49% r? EEK]: 16D£z3 4‘ 2 ’37. r 1. M37. '
'=' ‘$8.EZ — LL62 (“Ch—a)
d) There is about a 95% chance that X will fall in what range?
Oran, —> 53;] + 2.51331
: '52“: 25156.32
2 {plLMEi —— Maw (45 — I0 e) There is about a 99.7% chance that X will fall in what range? ~ _ 61172—9 EEXE’F'ﬁ‘SD [31 E
: ~$Zi3~$632 ? 2 “5220.616; A $6.36: (’21 “17). 4’) What Joe? {Li‘nk abou‘é {APSR fan (as? L§¥e§w£qts flanks they‘re
Pfe‘tty 3066’, 69 Example 16: Lemonade Stand Little Lisa owns a lemonade stand and sells lemonade on the corner of Lemon Lane
and Ade Avenue. Lisa always sells between 7 and 11 (inclusive) glasses of
lemonade per day. The daily sales, X, of glasses of lemonade follows the
probability distribution below: (a) (b) (C) (d) (e) x 7 8 9 10 11 V
P(X=x) 0.15 0.25 0.25 0.10 0‘25 What is‘ the missing probability that Little Lisa sells 11 glasses of lemonade
on a given day, P(X = 11)? l' 0.l5o.15—0.’Z_5—OIO =EE What is the probability that Little Lisa sells at least 9 glasses of lemonade on
a given day, P(X 2 9)? 0.25 + 0.10 + 0.15 : What is expected number of glasses of lemonade Little Lisa sells in a given
day, ElX]? ﬂmons + «3.0.25 + Cl 0.2.6 + 100.10 + H015 : 7: foCXTJc) = ___ [ED—“EA
What is the standard deviation of g asses o emonade Little Lisa sells in a
given da SD X]?
: Vicar. P 0w) :
r In 8 weeks (56 days), Little Lisa is expected to Sell around 50498
glasses of lemonade, give or take (0.43:3 glasses or so. Elfsmlr WEEK—L = 54,. €1.05 :
5DE5‘M1: M'ﬁbe] = l'3‘l5‘5 =(10.q%5) l
l
i
l l
i x
x 'L
(74.05)? 015 + (8—9051015 +(ac1.os)~o.25l
'2 l + (104.0532010 + (ll—aloe) o.7J—3 Example 17: Poker Hand
C—Meck is an awesome poker player and he loves to play 7Card Draw. Let X denote the number 0f clubs dealt. to C—Meck in a seven—card hand. (a) (b) L (C) Create the box model to represent this situation. [:13 [33. 3a [£7] {P57 w/o replacement . $01 =5uM Compute the. average and standard deviation of the box. _ 'l3l+'5“lQ’ _L _, Ill3'3ﬂ Iv
Abox ’ 52 f = H 6:30: " [1‘01 52" “43453 C—Meck is expected to'b’e dealt about v l ~ 7 5 0 clubs in a sevencard hand,
give or take l Q 7 (9 clubs or so. Example 18: Stat Student Work Hours A survey question asked two hundred STAT 100 students: “On average, how many
hours do you work each week?” The data collected was normally distributed with a
mean of 20 hours per week and a standard deviation of 2.5 hours per week. (a) Approximately what percentage of the students work, on average, at most
13 hours per week? P(X as): W243i?) = W2 <~ ’22) : Mow least 27 (b) Approximately what percentage of the students work, on average,
hours per week? 0 00 Zé P(X227)> P(Z 2 22.3% P622 263‘ = (c) Using the standard normal table, how many hours a week, on average, would
a student have to work in order to be at the 30th percentile? F(—X£1>:O'7’ 3) DC‘z‘>:—o.E>Z‘5
DLZO 2) 2.5 :C).?> 2.5 ::>(>L'Z (9.6g75> (d) Suppose a simple random sample of 65 of these students is chosen. You
would expect the mean average work hours per week for these students to begabout 2 Q hours, give ortake (9.2555 hours or so.
13le : Mb” = 20 ~ ,— 6_t>:> N‘n __ 2'5 200'45 ”
5‘)le ’ \rﬁx ’ {g}; ‘VZoo'i “0255* Example 19: Multiple Choice Test A multiple choice test has 75 questions and each question has five choices, oniy
one of which is right. Each right answer is worth 4 points and, to discourage
guessing, each wrong answer is worth 2 points. Peter has had no time to study for
this test and randomly guesses on each question. (a) Set up a box model to represent Peter’s total score on this test. ELL E3 ‘4 (1:75 w/rcplqccmcné. SCI =$um
(b) Peter is expected to score about " Q20 points, give or take 20 78L’I6 l (c) Is randomly guessing on a question a good idea for this test?
N0) V“r+ua\l\/ 53cm C't’idu’lcf mg hon/m? (JoS it"x/e
ou'eCon’M‘ (Foér’fh/g £€§"ﬁ $Cofe 3 ' ...
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This note was uploaded on 08/08/2009 for the course STAT 11 taught by Professor Hirtz during the Spring '09 term at University of Illinois at Urbana–Champaign.
 Spring '09
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