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3C Practice Problems for Midterm 1 Sudesh Kalyanswamy (1) Bill’s school schedule has four timeslots. Bill has 30 choices of classes for the first timeslot, 20 for the second, 23 for the third, and 17 for the last. How many possible schedules are there? (2) Consider the letters a, b, c, d, e, f , and g. (a) How many 7 letter words can be formed from these letters if letters may not be repeated? (b) How many 4 letter words can be formed from these letters if letters may not be repeated? (c) How many 6 letter words can be formed if letters can be repeated? (d) How many 6 letter words which begin with a and end with g are there if letters may not be repeated? (e) How many 6 letter words which begin with a and end with g are there if letters may be repeated? (f) How many 5 letter words can be formed which have at least one “c” if letters may not be repeated? (g) How many 5 letter words can be formed which have at least two c’s if letters may be repeated? (3) How many 8 letter words can be formed using 3 A’s, 2 B’s, 2 C’s and 1 D? (4) Three committees of three are to be chosen from a group of 11. How many possible committees can be chosen if: (a) The committees are assigned to different tasks and no person can be on more than one committee? (b) The committees are assigned to different tasks and people can serve on multiple committees? (c) The committees are assigned to the same task and no person can serve on more than one committee? (5) Six cards are drawn at random from a standard deck of cards. How many ways are there of drawing 3 kings and 3 aces? (6) Alice is taking a multiple choice test. There are 10 questions, each with 4 possible options. In how many ways can Alice answer the questions if: (a) Alice must answer all the questions? (b) Alice has the option of leaving questions blank? (c) Alice may leave no more than 2 questions blank? (7) Eight horses are entered in a race. (a) How many different possible orderings are there for completing the race (assuming no ties)? (b) How many different ways are there of distributing a gold, silver, and bronze medal (again assuming no ties)? (8) Telephone numbers consist of 7 digits, but the first digit cannot be 0 or 1. If digits can be repeated, how many different telephone numbers are there? 1 (9) A restuarant offers 8 main courses, 7 drinks, 5 side dishes, and 8 desserts. You decide to take a special which allows you to have 2 side dishes. How many different meals consisting of one main course, one drink, two sides, and one dessert are there if: (a) Your choices of sides must be distinct? (b) You are allowed to order the same side twice? (10) A certain password requires 10 digits, but the first seven digits must be between 0 and 6. How many passwords are possible if numbers can be repeated? (11) A palindrome is a string of letters which reads the same forward and backward (example: ABCBA). How many different 5 letter palindromes are possible (letters can be repeated)? (12) A group of four men and four women are to be seated in a row for a photograph. How many ways can this be done if: (a) The women will sit together and the men will sit together? (b) The seating will alternate gender (i.e. man, woman, man, woman, etc.)? (13) A committee of seven is to be chosen from a group of 20 candidates. The committee has positions of a chairman, vice chairman, and secretary, but the other four spots have no titles. How many different committees can be formed? (14) There are twelve people at a company picnic. They are going to line up for a photograph. How many different possible pictures are there if: (a) Alice and Bill want to stand next to each other? (b) Alice and Bill refuse to stand next to each other? (15) In how many ways can four pennies, three nickels, two dimes, and one quarter be arranged in a row? (16) There is a large house which needs cleaning. There are 8 workers. Five are needed to clean the carpets, two are needed to clean windows, and one is needed for the rest of the house. In how many different ways can the tasks be assigned? (17) A team of nine is to be chosen from a group of 11. However, two of the people in this group refuse to be on the same team, so can’t both be on the team together. How many different teams can be formed? (18) A number from 100 to 999 (inclusive) is chosen randomly. What’s the probability that at least one digit is repeated? (19) A license plate consists of three digits followed by three letters, where characters can be repeated. A license plate number is chosen at random. What’s the probability that: (a) The first digit is a 1? (b) At least one of the digits is a 0? (c) The first digit is a 1 and the first letter is an E? (d) The first digit is a 1 or the first letter is an E? (20) A fair die is tossed twice in a row and the numbers are recorded. (a) What’s the sample space of this experiment? (b) What’s the probability that the first number is a 2? (c) What’s the probability that the sum of the numbers is odd? (d) What’s the probability that the product of the numbers is odd? 2 (e) What’s the probability that the product of the numbers is even? (f) What’s the probability that the maximum of the two numbers is at most 3? (g) What’s the probability that the difference between the two rolls is exactly 1? (21) A true/false test has 10 questions. You take the test and guess on all the problems. What’s the probability that you get at least 8 problems right? (22) A card is drawn at random from a standard deck. What’s the probability that it is a 9 or a spade? (23) 6 cards are drawn without replacement from a standard deck. What’s the probability that: (a) You get a three of a kind, a pair, and a separate single value? (b) You get two sets of three of a kind? (c) You get three pairs (each with a different card value)? (24) There are five red, six green, and four yellow balls in an urn. You draw two balls from the urn. What’s the probability that they are of the same color if the balls are drawn: (a) Without replacement? (b) With replacement? (25) A subset of 4 elements is randomly chosen from the set {1, 2, 3, 4, 5, 6, 7}. What’s the probability that the subset contains the number 5 or the number 7? (26) A word is formed randomly from 2 A’s, 2 B’s, 2 C’s, 1 D, and 1 E. What’s the probability that the word begins and ends with A? (27) Alice and Bill have five children. What’s the probability that they have at least 2 boys? (28) A fair coin is tossed 10 times. What’s the probability that there are exactly 5 heads and the first and last toss are both heads? (29) What’s the probability of getting the following poker hands: (a) Single pair (b) Two pair (c) Three of a kind (three cards of the same value and two cards whose value is different from the set of three and from each other) (d) Full house (three of a kind and a pair) (e) Four of a kind (four cards of the same value) (f) A straight (five cards of consecutive value, like 34567) (g) A flush (five cards of the same suit) (30) Alice, Bob, Cindy, Dave, and Eddie are seated randomly at a round table. What’s the probability that Alice is not sitting next to Bob? (31) Ten cards are drawn from a standard deck. What’s the probability of getting exactly 4 9’s or 4 10’s if the cards are drawn without replacement? (32) In a given school, there are 60 students in either band or orchestra. There are 35 students in band and 40 students in orchestra. You pick a student at random. If you know that they are in the orchestra, what’s the probability that they are in band? (33) You are playing monopoly and you roll a pair of dice. Unfortunately, one went under the couch so you can’t see it. You can see that the other roll was a 4. What’s the probability that the sum of the rolls is at least 9? 3 (34) A fair coin is tossed twice. Let A be the event that the first toss is a heads and B the event that both tosses are the same. Are A and B independent? Justify completely. (35) A fair coin is tossed 10 times. You know that the fifth toss is a head. What’s the probability that, in the 10 tosses, there are exactly 5 heads? (36) A true/false test consists of four problems. A student takes the test by guessing each answer. You are told the student gets at least one right. What’s the probability that the student: (a) Got all four questions correct? (b) Got at least three questions correct? (c) Got exactly one question correct? (d) Got exactly two questions correct? (e) Got at most one question correct? (37) There are two urns consisting of different colored balls. Urn 1 has 10% red balls, 40% yellow balls, and 50% green balls. Urn 2 has 10% red balls, 60% yellow balls, and 30% green balls. A fair coin is flipped. If heads, you pick a ball from urn 1, and if tails, you pick a ball from urn 2. (a) What’s the probability that you pick a red ball? (b) What’s the probability of getting a yellow ball? (c) What’s the probability of getting a green ball? (d) Given that the coin comes up heads, what’s the probability of picking a yellow ball? (e) If A is the event that you pick a yellow ball and B the event the coin comes up tails, are A and B independent? Explain. (f) If A is the event you pick a red ball and B the event the coin comes up heads, are A and B independent? Explain. (g) Given that you pick a green ball, what’s the probability that the coin came up tails? (38) A certain game is played as follows: There are 10 closed boxes, and 2 of them contain a prize. You pick two boxes, and if there is a prize in both boxes you win. Otherwise, you lose. What’s the probability that you win the game? (39) There is a deadly disease which affects 1 in 100 people. A lab test can correctly identify positive cases 99% of the time and correctly identify negative cases 95% of the time. A randomly selected person tests positive. What’s the probability that they have the disease? (40) There are two boxes, each containing colored balls. The first box contains 2 red and 3 green balls, and the second box contains 4 red and 1 green ball. A random ball is chosen from the first box and placed into the second box. A ball is then chosen from the second box. (a) What’s the probability that this second ball is green? (b) Given that the second ball is green, what’s the probability that the first ball was also green? 4 Answers: (1) 30 · 20 · 23 · 17 (2) (a) 7! = P (7, 7) (b) P (7, 4) (c) 76 (d) P (5, 4) (e) 74 (f) P (7, 5) − P (6, 5) (g) 75 − 65 − 5 · 64 (3) 8! 3!2!2! (4) (a) (b) (c) (5) 43 · 11 3 8 5 3 3 11 3 3 1 11 8 5 3! 3 3 3 4 3 (6) (a) 410 (b) 510 (c) 410 + 10 · 49 + 10 2 · 48 (7) (a) 8! (b) P (8, 3) (8) 8 · 106 (9) (a) 8 · 7 · 5 2 2 ·8 (b) 8 · 7 · 5 · 8 (10) 77 · 103 (11) 263 (12) (a) 2 · 4! · 4! (b) 2 · 4! · 4! (13) P (20, 3) · 17 4 (14) (a) 2 · 11! (b) 12! − 2 · 11! (15) 10! 4!3!2!1! (16) 8 5 3 2 (17) 9 − · 11 · 11 9 7 (18) 1 − 9·9·8 900 (19) (a) 1 10 (b) 1 − (c) 9·9·9 10·10·10 1 260 5 (d) 1 10 1 26 + − 1 260 (20) (a) Pairs of numbers where each number is between 1 and 6. (b) 1 6 (c) 1/2 (d) 1/4 (e) 3/4 (f) 1/4 (g) 10/36 (21) (22) 10 10 (10 8 )+( 9 )+(10) 210 1 13 + (23) (a) 1 4 − 1 52 13·12·11·(43)·(42)·(41) (52 6) 2 (b) 4 (13 2 )·(3) ) (52 6 (c) 4 (13 3 )·(2) 52 (6) 3 (24) (a) (b) (52)+(62)+(42) (15 2) 52 +62 +42 152 (25) (63)+(63)−(52) (74) (26) 6!2! 8! (27) 32−1−(51) 32 (28) 210 (83) 3 (29) (a) (b) (c) (d) (e) 4 4 13·(12 3 )·(2)·(1) 52 5 ( ) 13 2 ( 2 )·11·(42) ·(41) 52 5 ( ) 12 2 )·(43)·(41) (52 5) 13·( 2 13·12·(43)·(42) (52 5) 13·12·(44)·(41) (52 5) 5 (f) (g) 10·(41) 52 5 ( ) 4·(13 5) (52 5) (30) 1/2 (31) 4 48 4 4 44 (44)·(48 6 )+(4)·( 6 )−(4)(4)·( 2 ) 52 (10) 6 (32) 15/40 (33) 1/3 (34) Yes, since P (A ∩ B) = 14 , P (A) = (35) 1 2 and P (B) = 12 . C(9,4) 29 (36) (a) 1/15 (b) 1/3 (c) 4/15 (d) 6/15 (d) 4/15 (37) (a) .1 (b) .5 (c) .4 (d) .4 (e) No, P (A ∩ B) 6= P (A)P (B) (f) Yes (g) (.3)(.5) .4 (38) 2/90 (39) (.99)(.01) (.99)(.01)+(.05)(.99) (40) (a) (b) 4 15 3 4 7