midterm-solutions

midterm-solutions - Test 03/03/201 1 Name 1. Consider the...

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Unformatted text preview: Test 03/03/201 1 Name 1. Consider the following experiments. In each case, provide the information requested. (a) [3pts] A coin is ﬂipped 3 times An example of a typical outcome is: HHT The total number of outcomes in the Whole sample space is: g (b) [3pts] Three dice of different colors are rolled A) ' ' 5’ An example of a typical outcome is: 6 m 5 The total number of outcomes in the Whole sample space is: 5 ﬂag; Emmiésﬁ) (c) [3pts] A card is drawn from a deck, and then without replacing it, another card is drawn An example of a typical outcome is: 2 43) The total number of outcomes in the whoie sample space is: 52 ’57 (Cl) [Spts] A card is drawn at random from a deck; it is replaced, and then another card is drawn An example of a typical outcome that is not in the sample space of (c) is: f 3 ﬁg 3H) The total number of outcomes in the whole sample space is: 52 ’ 5—2 . 2. A certain experiment has a sample space containing 100 different outcomes that are all equally probable. Event A contains 50 outcomes and event B contains 40 outcomes. (a) [Spts] If A and B have 30 outcomes in common, what is the probability that neither A nor B occurs? 2/5— {b) [3pts] If A and B are independent, what is the probability that both A and B occurs? ’27; (c) [3pts If A and B are mutually exclusive, What is the probability that either A or B occurs? (/0 3. Three letters (possibly with repeats) are selected at random from the alphabet and written in the order selected. What are the following probabilities? Your answer may be an expression such as 26 - 25 - 24, and it may include symbols such as A vowel is one of the lettersnz, e, i, o, n. A consonant is one the remaining 21 letters. (5)3 (b) [3pts] P(the ﬁrst and last letters are consonants and the middle one is a vowel) < 3;. > (2% > (Continued...) (a) [3pts] P(all the letters are vowels) “l I. ('0 (\J (c) [Bpts] P(three diﬁ'erent letters are written) E 3i 26)(;é (d) [Spts] P (one of the three letters appears exactly two times, eg. “AAB”, “ABA”, etc.) #ow’rwmes {W = 2-5 (aka/texﬁrrwmfd/gﬁw) v 2516'wa tr sing/é latter w 3(ww54b Wée ﬂu/MWJ} =.. 329%. Sea) (e) [3pts] P(there is at least one vowel among the letters written) ’P(“) =_ 2&125'3 ,g . \ __ < 2.; 3 _ i; 5 a - 2-6 46““ (f) [3pts] P(three dtﬂerent vowels are written, GIVEN THAT only vowels were chosen) 5'; A; :3 m— S . 5-- t3-». f o (71 g (g) [3pts] P(the letters are in alphabetical order, GIVEN THAT all the letters are different) l/é? ( H? ﬂares: diffwMW-c 5W W ‘5 / W a “(paw/72M). 4. These problems concern the binomial distribution P(X=:L'ln_,p) = (:)pm(1—p)”“m, \$=O,1,...,n. (a) [4pts] Explain the meaning of the above. What is X ? What is :12? What is n? What is p? X is m v:anle oil: sarcasm is )7 Mwa 766% ,‘,£ 7%: Frat. ab may: 9'” W M :1! f. X J‘s m Pat/9955mm VM'M M 4d 1‘5“ 0%! at? f]? ya59}/e VW‘ (b) [4pts] If a fair coin is ﬂipped 6 times, what is the probability that at most two heads are obtained? (tit/aﬁ- mic/zit (my: (C) [4pm] If a fair die is rolled 6 times, what is the probability that at most two sixes are obtained? l3><%)0(2)6+(?)<’/6)(%)i (gxmgﬁ: 5,55 éé 5. Two dice are rolled. Let X be the total rolled (i.e., the total number of dots showing on the up faces). Let A be the event that at least one of the dice comes up on a number that is greater than or equal to 4. Let B be the event that both dice land on numbers less than or equal to 4. Let C be the event “X is divisible by 4.” (a) [5pts] Are A and B independent? Why or Why not? WMPCB): 'é We wept game) e 77% (c) [Epts] Are B and 0 independent? Why or Why not? a it; :5 s \ P<QPC®~ at Q ﬁg: We? ﬂtnclzh 6. [6pts] Two dice are rolled. Let W be the product of the numbers showing (e.g., if One die lands with 3 up and the other with 5, then W = 15). Find A f“! @525 : 2—1-2; EM): 3: any}; {glen vg/é: i:{ J“ n "-u' -._\] 4: 4» r». + .._.'I: (79 (9; 7. [lﬂpts] The test for a cigain disease is positive 98% of the time if given to a person with the Q disease, and it ' - e time if given to someone who does not have the disease. lOnly . one in a hundred have the disease If a random person tests positive, what it the probability she he has the 1sease? Test. * 13(F’osrtesi' i Dime-u) ?(Dikm) : F’Mi’ I viewﬁtww) 4,. T’Q‘Pshs-r “90%) _ 8.An experiment has three possible outcomes with probabilities a, b and c, respectively. (Thus @ij 119 a + b+ c = 1.) The experiment is performed three times, with each trial independent of the others. (a) [Spts] What is the probability that each of the ogétcomes occurs once? 4; “ll/awe M S’i‘wluﬂ) (“99°02 (q/a/ﬂk 6+6 ' Raolgﬁb (Kmin (Raﬁ); (65232“); (3:59)) (K) ‘ ' e9 «hm r96, GLbc. S (b) [Spts] What is the probability that alighkrgle‘ 0% cocrgggx’aifg iifé wt? 99 0 same' _ mar): gate, m E 2 E (ii/99%),- Cﬁ; is», p); (at a; U} . 35% ' m. 3 3 .3 13s ’7 _. e. \(E), gadget 2” . Q. Q 9. [Spts] There are 3 white and 3 black balls in a box. Two are taken at random] removed and NE E replaced by black balls, then 2 are selected. What is the probability that the selected balls are the same color? ...
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This note was uploaded on 11/29/2011 for the course MATH 3355 taught by Professor Britt during the Spring '08 term at LSU.

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midterm-solutions - Test 03/03/201 1 Name 1. Consider the...

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