Exam_3_Questions

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© COT 3100.001 Spring 2022 Exam 3 (Final) Name: Q o “4 }iony U-Number: Instructions 0. £ cullies” @J @, @, @ /g:g::i Read all instructions carefully. In order to receive full credit, correct solutions and answers C 4 must be well-organized and printed legibly, following the given instructions. Please only O MKkt place final answers for the questions and parts in the blanks, using the space outside and o. scratch paper to perform any intermediate calculations. There are 120 points on this exam 'S Cf'a.'l‘cl\ but it will be graded out of 100. pa' 3¢ 1. (6 points) Let A and B be any events in some general sample space §. Match each expression i. through iii. to an equivalent expression or expressions from choices a. through h. Note that some choices may be valid only for particular types of events A and B, but we are looking for expressions for which the equalities always hold. List the letter choice(s) in the corresponding boxes i. P(ANB)= ii. P(AUB)= iii. P(S)= a. P(A)+ P(B) e. P(B)-P(A| B) b. P(A)+ P(B) — P(AN B) f. P(A)+ P(A°) c. P(A)-P(B) g. P(B) + P(B°) d. P(A)- P(B| A) h. 1 (one) 2. (5 points) Assuming that for Question 1 above, each of the parts i. through iii. has at least one matching answer from expressions a. through h., how many ways are there to guess at answers to the three parts of Question 17 Hint: Counting the non-empty subsets of {a,b,...,h} is part of the solution. Answer:

COT 3100.001 Spring 2022 Exam 3 (Final) that is in your pocket. You flip the coin five times and it comes up tails each time. After a thorough inspection, you are confident that the coin has two distinct sides, heads and tails, and is perfectly balanced. The probability that the next flip of the coin results in tails is: @ 3. (4 points) You are bored one day waiting for class to start and you begin flipping a coin Exactly one half. Less than one half. More than one half. 4. (6 points) A class of 233 students meets twice a week for lecture, after which they separate into eight breakout groups for discussion. What is the smallest number of students that can be in the largest breakout group? Answer: all balls of one color have been removed, and then we stop drawing. On average, how many balls will be remaining in the urn once we stop? Give an exact answer or round your answer to four decimal places. @ 5. (8 points) An urn contains 2 red balls and 2 blue balls. Balls are drawn randomly until Answer: N

COT 3100.001 Spring 2022 Exam 3 (Final) 6. (4 points) Count the following;: The number of ways in which a (a) coin can be flipped 10 times and land on heads exactly 5 times. (b) The number of 5-element subsets of a 10-element set. () The number of arrangements ®) of ABABABABAB. The coefficient of p°¢® in the (d) - 10 expansion of (p + ¢)™. 7. (6 points) Five women (W) and five men (M) line up at a checkout counter. What is the exact probability, expressed as a fraction, that they are arranged from first to last as W. M, W, M, W, M, W. M, W, M? Assume that all possible arrangements are equally likely. The exact probability is 8. (6 points) A fair coin is flipped ten times. What is the exact probability, expressed as @ a fraction, that the coin comes up tails exactly five times? The exact probability is Page 3

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COT 3100.001 Spring 2022 Exam 3 (Final) @ s o " 11. ® (6 points) What is the constant term (where all z’s cancel) in the expansion of (2z + %) %9 Answer: (4 points) A coin is weighted so that heads comes up twice as often as tails. What is the expected number of heads in 72 tosses? Answer: (5 points) How many solutions are there to a+b+c+d+e=30 where each variable must be a nonnegative integer? Answer: . (5 points) How many solutions are there to a+b+ct+d+e<30 where each variable must be a nonnegative integer? Hint: The number of solutions to a + b+ c+d+e = k for each 0 < k < 30 (k is constant) is the same as the number of solutions to a + b+ c+d+e+ f = k+ f where k+ f = 30. Answer: Page 4

COT 3100.001 Spring 2022 Exam 3 (Final) number is as likely to be chosen as any other number. Determine the exact probability, a fraction, that the randomly chosen number is a multiple of 5, 7, or 11. The fact that there are |_§J integers from 1 to b that are multiples of by d will be helpful. The blanks below should give you an idea of how to proceed. @ 13. (10 points) A random number from 1 to and including 12,345 is chosen, where each | Ms| + | M7| The exact probability is + ‘M 11| 8, and ten six-sided dice which are blue, with sides numbered 1 through 6. The dice are perfectly balanced, i.e. each number on each of the dice is as likely to come up as any other number. You roll all twenty dice and then take the sum of the red dice and then subtract the sum of the blue dice. @ 14. (5 points) You have ten eight-sided dice which are red, with sides numbered 1 through What is the expected result of this experiment? Round your answer to four decimal places. Answer: \- Page 5

COT 3100.001 Spring 2022 Exam 3 (Final) 15. (10 points) Martian Poker is a game played where each player receives four cards from a deck of 119 cards, where there are 7 suits and 17 cards from each suit. The order in which the cards are received does not matter. Answer: : (a) How many possible four card Martian Poker hands are there? (b) A flush in Martian poker is any hand where all cards are the same suit. How many possible flushes are there in Martian Poker? Answer: (c) What is the probability of being dealt a flush in Martian Poker, expressed as a percentage? Round your percentage to four decimal places Answer: % Page 6

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COT 3100.001 Spring 2022 Exam 3 (Final) 16. (10 points) A discrete mathematics professor has fifteen students, each of whom can earn a grade of A, B, C, D, or F for the course. assign a letter grade for every student? Give a useful expression for this number (a) How many grade distributions are there? That is, how many ways are there to @ since your calculator will give it in scientific notation. Answer: grade, the number of students with that grade is the same as the number of students with any other grade. How many preferred distributions are there? You may again give a useful expression here rather than the number. Hint: AAABBBCCCDDDFFF is one example distribution. O (b) The discrete mathematics professor prefers distributions where, for each letter Answer: (c) Assuming all distributions are equally likely, what is the probability, expressed as a percentage, that the grade distribution for the course is a preferred one? Round @ your percentage to four decimal places. Answer: % Page 7

COT 3100.001 Spring 2022 Exam 3 (Final) ok (10 points) As a promotion, you are offered a free play at the following game: Three balls are drawn randomly, without replacement, from balls numbered 1 through 10. Players try to guess the three numbers drawn, but the order that they are drawn or guessed does not matter. e If the player matches all three numbers, they win $200. e If the player matches exactly two numbers, they win $10. e If the player matches exactly one number, they win $1. What can you expect to win playing this game in dollars on average? Round your answer to the nearest cent, i.e. to two decimal places. Answer: $ . (10 points) The U.S. Senate consists of 48 Democrats, 50 Republicans, and 2 Indepen- dents. A committee of five is to be formed by selecting some senators at random, where all senators have the same chance of being selected. What is the probability, expressed as a percentage, that the committee members all belong belong to the same political party? Round your percentage to four decimal places. Answer: % Page 8

COT 3100.001 Spring 2022 Exam 3 (Final) ina (Scratch Paper) Qj(C) Prob, aviomg sl results @ 2. (A ) T"'-'c!"}"“"’ S'l'h.;ql-""f\oruawi (‘0““" '\w, Q3(D) Gavallers Pollacy QCC) P79nul..’¢ Pr.‘qc.',ofe . e gs '(B) GXP'CJ'Z V;'M(. F:‘ul«”O‘\"'ﬂuc,ahJ $(e.v pv,L,A,I,"-. 5 Q6.(5, D,<.D) All theSame aus vty Somnne mere oLv:aw/aMomaJ;(ﬂ,,., Otlerr Q7(D.) P"PP’V QLS(&-VO\":DMJ 'L\’"’M Q é, bl‘wovﬂ.d( TFheortm Q, '(D) C}Y t\i&JS ar¢+u?c¢aswthvfii 2% +o:.,5 l'l'\72_ +°$Ses' @.,:(D) Sc,fec'L |.,/ ré P. V¢ LMV( geen '“\-'S Ane Ma‘«r‘f-‘v-\e} @,2 (A) Tr.‘c|</ +wi s+ on Qi bt L-wt previles a \ve Y toloolka? ;4. . S.’m,’hr'v/‘ola(gw.'u C(LS;J-[-\.,,,.W,‘_ ‘)“"'“" @,3.(5) IMC[hHO'l'Grc‘ws-'Wﬂ. - L p 5 / l- | - 0 CYaln, Olq.(C)L'“"" s CK’ écle -O"'. g\'fi"""‘rw\ovc déVthc/'{'lnalA‘ to-class ex~mple, Que, (DB D) Streigtiforcard applicalivns of Combruadions v (1, rale Proboblity. Q'B,(C)B,D) S.'(‘Vafgla‘{'poruar& appl_'w.'lfoh O": b"\U\’-,' V‘U\’PJ rronge vl ree Prokabslity Q'?, (D) (/alleed Tarovgl o n Qu’z 3. Slowld bave been (O freepointy, Wl taneepts relevent tlSevherc, @,‘.(B) APP‘? C'OIML.'ha.J.'auf l.// CLJ!.'IFO'-\ r‘U\’c or Pr°”“"'.l""" wl taaluclly exclas, ve events

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COT 3100.001 Spring 2022 Exam 3 (Final) (Scratch Paper)

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