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Unformatted text preview: Probability and Statistics for Engineering (ESE 326) Exam 1 February HZ 2011 This exam contains nine: multipienchoico problems worth two points each, three truewfalse probierns
worth one poﬁnt each, If} short—answer probioms worth one point cash} and two freoresponso probfans
worth nino points aitogether, for an exam total of 40 points‘ Part I. M'ultiQEeChoice (two points each) ClearEy ﬁll ﬁn the oval on your answer card which corresponds to the 0111}; correct response. 1. Each time Acme Suppiies lands a new customer, an eightucharacter alphanumeric code for that
customer i3 generated randomiy by a computer. Each character in such a password can be chosen
from among the 26 letters AWZ and the 10 digits 0%). What is the probabitity that a randomiy«
generated password of this type contains no repeated letters or numbers? (A) .000011
(B) 009025
(C) .2523
(D) .3016
(E) .4325
(F) .5675
(o) .6984:
(H) .747?
(I) .9999?5
(5) 399989 ix) In how many ways can six peopio be chosen from a pool of seven females and nine males, where at
least one female and one male must be included? (A) 54 (:8) 91 (C) 189 (D) 882 (E) 10?: (F) 1260 {(3) WE”? (H) 8008 (I) 6&363
{3) 1,513,512 3. What is the probabiiity that a sevewcard hand drama from 31 Strander deck of S2 oards contains one
throovof—akind and two (different) pairs? (An oxampie of a hand like this would bo
ShSaﬁQOQoQﬁQv.) (For you poker aﬁcionados, I undorstmzd that the second pair is
irrelevant since only ﬁve of {he cards can be utiiizod, Emit humor me: and compute the. probabiliiy
anyway.) (A) .0803 (B) .0604 (C) .0009 (D) .0012 (E) .0018 (F) .0024 (G) .0074 (H) .0210 (I) .0222 (J) .0443 28% of American households own a dog but not a cat, 15% own a eat but not a dog, and 51% own
either a cat or a dog or both. What percent of Amoﬁcan households own both a dog and a cat? (A) 7%
(B) 8%
(C) 12%
(o) 13%
(E) 16%
(F) 20%
(o) 21%
(H) 34%
(I) 41%
(.3) 49% L11 Suppose A and B are evente such that PR4} m .4, 2 .275, and PERM} m .6. Find (A) .696
(B) .14.;
(C) .18
(D) .24
(E) .3
(F) .32
(G) .45
(H) .5333
(I) .6777
(3) .8 Consider the population of American males ages 18 to 65. 70% of those who are 18 to 30 years oid
(“younger guys”) are employed, 92% of those who are 31 to 50 years old (“middle—aged guys”) are
employed, and 86% of those who are 51 to 65 years oid (“oider guys”) are employed. Within this
whole population, 25% are younger guys, 60% are middlewaged guys, and 15% are cider guys. If
an American male 18 to 65 years old is emoloyed, What is the probability that he is an older guy?
(A) .1507 (B) .1578 (C) .1656 (B) .171? (E) .1789 (F) .1950 (G) 32—04% (H) .2632 (1) .3551 (J) .6449 Grandpa E3011 heids an egg burn: for his grandchiédren every Easter. He puts a quarter in each 01“ {$0
eggs and puts a doﬂar biﬁ in each of eight eggs. H“ Kim ﬁnds nine eggs this year, What is the
probability mat She gets exactiy {we of the eggs wﬁh doifar bﬂls? {A} .003:
(B) .8086
(C) .9111
(D) .0641
(E) .1359
(F) .24?2
(G) .279:
(H) .3113
(I) .3447
(J) .3662 Cowboy Bob ﬂips a coin every day to determine Whether or not he wili take a bath that day‘ If he
gets heads! he takes a bath; if he gets tails, he does not. The problem is that he uses a weighted
coin (given to him by his granddaddy, Gambler Gus) which comes up heads only 40% of the time.
If Cowboy Bob considers Sunday to be the first day of the week, Wha’é is the probability that his
first bath of the week will be on Tuesday or Wednesday? (A) .0864
(B) .1344
(C) .1440
(D) .2304
(E) .3840
(F) .6160
(G) .’?696
(H) .8560
(1.) .8656
(3') .9136 9. There are 295 sindenis enmiied in an intreéuctery anthrepoiogy course Of theee, 200 are aemaiiy
in the class, and ﬁve are on {he waétlisi, Suppose there is a 2% chance the: any given student
among the 2% wili withdraw eariy eneugh in the semegter t0 aiiow someene frem the waitiist te be
admitted inte the ciass. (You may assume that wiéhdrawals are indepemieni of one aﬁether, and
you may also assume that me one from the waitiist withdraws.) What is the probability that all the
students on the waitiist wiil get into the (23353? (A) .1035
(B) .2133
(C) .2815
(D) .3?12
(E) .4906
(F) .5994
(G) .6288
(H) .7185
(I) .7857
(I) .8914 93:111. True—False (one point each)
Mark “A” on your answer card ifthe statement is true; mark “B” if it is faise. 16. For any events A and B, PEAIB’] m 1 — PMIB]. 11. Let F be the cumuﬁative distributien function for a discrete random variable X. if x; : 332$ then 12. Let X be a discrete rande variabie. Then fer any real number 6, 06X m 50X. Part {I}. Short Answer (one point each) The answer to each of these is tight or wrong: no work is required. and no partial credit will be given.
Give only one answer to each. (if you give more than one answer. the poorer one will count.) For any
problem requiring a numerical answer, give a numerical answer, not just a formula or an unﬁnished
computation. 13. What single adjective describes a phenomenon fut which any individual result is unpredictable, but
a pattern emerges in the long run? 14. and 15. Let A and B be events such that PM} K .3 and P {B} m .5. Exactly two ofthe following
statements about A and B are true. Which ones are they? (I) If A and B are mutually exclusive, then PM U B] = .8.
(II) if A and B are mutually exclusive, then PEA R B] m .8.
(1H) 11“ A and B are mutually exclusive, then PEA U B] : .15.
(IV) If A and B are mutually exclusive, then P D B] m .15
(V) If A and B are independent, then PM U m .8.
(V1) If A and B are independent, then PM ﬂ BE :— .8.
(VII) If A and B are independent, then P [A U B} m .35.
(VIII) If A and B are independent, then PEA P. B} x .35. For problems 16 and 17, suppose X £5 a discrete random variable with y: t: 50 and 0’ x 2. 16. Find Var(3X + d). W. Fill in the blank:
According to C'hebyshev's Inequality, P [:12 < X < 58} 18. In two words, the Moment Generating Function is useful because it 19. Below is the derisity for a binomial random variable with n :2 40 and p 2 .3, except that the values
to which the formula applies are missing. Fill their; in. ﬂat) x (fﬂewmw i m 20. Given a Bernoulli trial with constant probability of success p = .16, what is the average number (3f
trials needed to obtain two successes? 21. Suppose theie are 66 marbles in a bag, of which 25 are blue. What is the average number of blue
maxbles in a sample of size 18 taken from the bag? 22. Suppose X is a hypergeometrie tandem variable with N = 1609, T x 50, ancl n m 20. Since N is
large relative to e, the binomial distribution may be used it) approximate the hypergeometric
distribution. What value of p sheuld be used? I’art W. Free Response (paint values as shown) FGHOW directions carefuliy, and Show 311 the steps needed {a arrive at your soiuﬁon, If you wish it)
1‘9qu any numbers, mqu 1:0 four decimai places. (6) 23. Consider the discrete random variabie X Whase density is given by the £0110“;ng Labia. 2:  1 2 4 8 16
.809 .8009 .8060?) (3) Find EiX (13) Find Var(X (c) Find 0X. ./ t\r
(3) 34 Fiﬂé the derivative 0f TEX“) :2 with respect t0 '5. You do not need to Simpkify your answer, 110: ($0 you need is plug in anything for i. ...
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