Mathematical Stat I: solutions of homework 1

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1 Mathematical Stat I: solutios of homework Name: Studet Id N:. Suppose we tur over cards simultaeously from two well shuffled decks of ordiary playig cards. We say we obtai a exact match o a particular tur if the same card appears from each deck; for example, the quee of spades agaist the quee of spades. Let equal the probability of at least oe exact match. (a) Show that = -! +! -! + -! Hit : Let C deote the evet of a exact match o the ith tur. The = P(C C C. Now use the geeral iclusio-exclusio formula give by(.3.4). I this regard ote that: P(C )=/5 ad hece PC = 5(/5)=. Also P(C C 50!/5!. (b) Show that is approximately equal to e = (sol of a) Theorem: For example, i i ( ) i, i a... a ai i i aa... ai P C S S P C C C i i k i i k If =, ( ) i P C S, S PC, S PC C i If =3, ( ) i P C S, S PC i i k i k, S P C C P C C P C C S P C C C It ca be show by the mathematical iductio!! For our questio, = 5 ad

2 5! 5! 5 50! P C C 5! PCi, i j P C C... C a a a i As a result, S PC i 49! P C C C,, 5!, i j k (5 i)! 5! i ! 5! 50! S PCi Cj i j5 5!!50! 5! 5 (5 i)! Si PCa C... a C a i aa... a 5! i i 5! (5 i)! (5 i)! i! 5! i! ad i pm P Ci ( ) Si... i i! 3! 5! (sol of b) By the Taylor theorem, e x e ( ) ( ) i0 i! i0i! ii! ( ) i pm e i i! i x i i. There are 5 red chips ad 3 blue chips i a bowl. The red chips are umbered,, 3, 4, 5, respectively, ad the blue chips are umbered,, 3 respectively. If chips are to be draw at radom ad without replacemet, fid the probability that these chips have either the same umber or the same color.

3 I a lot of 50 light bulbs, there are bad bulbs. A ispector examies 5 bulbs, which are selected at radom ad without replacemet. (a) Fid the probability of at least defective bulb amog the 5. (b) How may bulbs should be examied so that the probability of fidig at least bad bulb exceeds? (sol of a) Let X be the umber of defective bulb amog the 5. The, PX ( ) PX ( 0) (sol of b) If we select bulbs, the probability is 48 PX ( ) PX ( 0) (49 )(50 ) (49 )(50 )

4 4. Suppose the experimet is to choose a real umber at radom i the iterval (0, ). For ay subiterval (a, b) (0,), it seems reasoable to assig the probability P[(a, b)] = b a;i.e., the probability of selectig the poit from a subiterval is directly proportioal to the legth of the subitervals ad show that P[{a}]=0, for all a (0,). For ay a (0,), if we let a a C a, a ( m0 ) a where,,... ad m 0, ( a) (i) a a C a, a (0,) ( m0 ) (ii) C C...: decreasig, (iii) i C i {} a (iv) a a PC ( ) ( m ) 0 By (i), (ii) ad (iii), a a P {} a P Cilim P( C) lim 0 im ( m0 ) 5. Assume that P(C C C )>0. Prove that P(C C C C ) = P(C P(C C ) P(C C C )P(C C C C ). 0 0, 0 because PC C C PC C PC 3 C C C C C C 3. Thus we ca cosider the followig coditioal prob: P C C C C 4 3 P C C C C 3 4 P C C C 3

5 P C C C 3 P C C C 3 P C C P C P C C C, PC ad PC PC C C C P C P C C P C C C P C C C C P C C P C C C P C C C C P C P C C P C C C 3 6. A boy foud a bicycle lock for which combiatio was ukow. The correct combiatio is a four-digit umber, d, d, d, d, where d,,,3,4, is selected from,, 3, 4, 5, 6, 7, ad 8. How may differet lock combiatios are possible with such a lock? It is equal to the umber of samplig with replacemet: A office furiture maufacturer that makes modular storage files offers its customers two choices for the base ad four choices for the top, ad the modular storage files come i five differet heights. The customer may choose ay combiatio of the five differet-sized modules so that the fiished file has a base, a top, ad oe, two, three, four, five, or six storage modules. (a) How may choices does the customer have if the completed file has four storage modules, a top, ad a base? The order i which the four modules are stacked is irrelevat. (b) I its advertisig, the maufacturer would like to use the umber of differet files that are possible-selectig oe of the two bases, oe of the four tops, ad the either oe, two, three, four, five, or six storage modules. The maufacturer may select ay combiatio of the five differet sizes, with the order of stackig irrelevat. What is the umber of possibilities? (sol of a) For the heights of four storage module, (i) all of four storage modules have the same height: (a, a, a, a)

6 5 5 (ii) four storage modules have two differet height: (a, b, b, b), (a, a, b, b), (a, a, a, b) 5 30 (iii) four storage modules have three differet height: (a, b, c, c), (a, b, b, c), (a, a, b, c) (iv) all of storage modules have differet height: (a, b, c, d) As a result, the umber of possibilities is (sol of b) Now we ca have oe, two,, or six storage modules: (i) Whe we have a sigle storage module, The total umber of case is (ii) Whe we have two storage modules, The total umber of case is (iii) Whe we have three storage modules, The total umber of case is (iv) Whe we have four storage modules,

7 The total umber of case is (v) Whe we have five storage modules, The total umber of case is (vi) Whe we have six storage modules, The total umber of case is As a result, the umber of possibilities is Let A ad A be the evets that a perso is left-eye domiat or right-eye domiat, respectively. Whe a perso folds his or her hads, let B ad B be the evets that the left thumb ad right thumb, respectively, are o top. A survey i oe statistics class yielded the followig table: B B Totals A 5 7 A Totals If a studet is selected radomly, fid the followig probabilities: (a) P(A B ), (b) P(A B ), (c) P(A B ), (d) P(B A ). (e) If the studets had their hads folded ad you hoped to select a righteye-domiat studet, would you select a right thumb o top or a left thumb o top studet? Why? (a) P A B

8 (b) PA B 5 P A B 9 (c) 9 P B A 3 (d) (e) PA B > P A B. Thus, we have better chace to select a right-eye- 6 domiat studet whe we select a left thumb o top. 9. A eight-team sigle-elimiatio touramet is set up as follows: A E B C wier F G D H For example, eight studets (called A-H) set up a touramet amog themselves. The top-listed studet i each bracket calls heads or tails whe his or her oppoet flips a coi. If the call is correct, the studet moves o to the ext bracket. (a) How may cois flips are required to determie the touramet wier? (b) What is the probability that you ca predict all of the wiers? (c) I NCAA Divisio I basketball, after a play-i game betwee the 64 th ad 65 th seeds. 64 teams participate i a sigle elimiatio touramet to determie the atioal champio. Cosiderig oly the remaiig 64 teams, how may games are required to determie the atioal champio? (d) Assume that for ay give game, either team has a equal chace of wiig.(that is probably ot true.) O page 43 of the March, 999, issue, Time, claimed that the mathematical odds of predictig all 63 NCAA games correctly is i 75 millio. Do you agree with this statemet? If ot, why ot? (a) We eed 7 games to determie the wier, which idicates 7 coi flips.

9 (b) The probability is 7 (c) Each touramet game elimiates a sigle team ad we eed to elimiate 63 teams. Thus, we eed 63 games. (d) The probability is 63. log0 63log ad log0 log The statemet is ot true. 0. Assume that a isurace compay kows the followig probabilities relatig to automobile accidets: Age of Driver Probability of Accidet Fractio of Compay s Isured Drivers A radomly selected driver from the compay s isured drivers has a accidet. What is the coditioal probability that the driver is i the 6-5 age group? (i) The probability of a accidet is (ii) The probability that the driver i the 6-6 age group has a accidet is Thus, by (i) ad (ii), the coditioal probability is

10 . Let S = ad defie the subsets Α, B, =,, of S as follows: Α = x, y ; 3 x6,0y, B = x, y ; x y. The fid the limits, A ad B, of Α ad B whe goes to ifiity. A A ( icreasig ) ad B B ( decreasig ) A A x y R x y (, ) ;3 6,0 ad B B (0,0). Let S = { x iteger; x00 } ad defie the evets A, B, ad C by: A = { x S; x is divisible by 7 } B = { x S; x = for some positive iteger } C = { x S; x }. Compute P(A), P(B), P(C), where P is the equally likely probability fuctio o the evets of S. Number of elemet i A is 8 ad Number of elemet i B is 63 ad Number of elemet i C is 9 ad 8 7 PA ( ) PB ( ) PC ( ) Let S be the umber of all outcomes whe flippig a fair coi four times ad let P be the uiform probability fuctio o the evets of S. Defie the evets A, B as follows:

11 A = { s S ; s cotais more T s tha H s }, B = { s S ; ay T i s precedes every H i s }. Compute the probabilities P(A), P(B). (i) If T is observed four times or three times, we observe more T s tha H s. Thus, PA ( ) (ii) We ca have four T s: (T, T, T, T) We ca have a sigle H ad three T s: (T, T, T, H) We ca have two H s ad two T s: (T, T, H, H) We ca have three H s ad a sigle T: (T, H, H, H) We ca have four H s: (H, H, H, H) Thus, 5 PB ( ) 5 6 4