Section 4.1. The Nature of Probability. Homework Continued

Section 4.3: Tree Diagrams and the Counting Principle Homework #1-19: Use the Counting Principle to answer the following questions. 1) If two dates are selected at random from the 365 days of the year and are listed in the order they were selected. a) Use the counting principle to determine the number of possible outcomes if the dates are selected with replacement. b) Use the counting principle to determine the number of possible outcomes if the dates are selected without replacement. 2) Two dates are selected at random from the month of May and listed in the order they were selected. a) Use the counting principle to determine the number of possible outcomes if the dates are selected with replacement. b) Use the counting principle to determine the number of possible outcomes if the dates are selected without replacement. 3) Joe has 5 shirts, 3 pairs of pants and 4 pairs of shoes. He needs to make an outfit containing one of each item. How many different outfits are possible? (We assume that one different item makes a different outfit) 4) Shannon is going to purchase a new car. The car has 5 exterior color choices, 2 interior color packages and 2 engine options. How many different cars are possible? 5) A television remote has buttons for the digits 0 – 9. If you press two buttons, how many numbers are possible if a) The same button may be pressed twice? b) The same button may not be pressed twice? 6) A safe’s lock has a three number combination. The numbers on the safe lock are 0 – 36. How many combinations are possible if no number may be used more than once? 7) A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased? 8) Girls' ice skates come with the following options: Colors: white, beige, pink, yellow, blue Sizes: 4, 5, 6, 7, 8 Extras: tassels, striped laces, bells Assuming that all skates are sold with ONE extra, how many possible arrangements exist? Section 4.3 Continued: 9) A state issues license plates consisting of letters and numbers. There are 26 letters and the letters may be repeated. There are 10 digits and the digits may be repeated. How many possible license plates can be issued with two letters followed by three numbers? 10) An ice cream shop offers 31 flavors. You order a double-scoop cone. In how many different ways can the clerk put the ice cream on the cone if you wanted two different flavors? 11) Burger Queen offers 4 types of burgers, 5 types of beverages, and 3 types of desserts. If a meal consists of 1 burger, one beverage and one dessert, how many possible meals can be chosen? 12) Elizabeth is choosing silverware with which to eat dinner. There are 3 knives and 2 forks to choose from. How many different silverware sets can Elizabeth choose? 13) Reid is ordering a birthday cake for a friend. There are 5 cake flavors and 4 frosting flavors to choose from. How many different cakes can Reid order? 14) If car license plates consists 3 letters followed by 3 numbers, how many license plates are possible if there are no restrictions? 15) Answer question 14 but assume the license plates must be of the form 2 letters followed by 4 numbers and there are no other restrictions. 16) The standard New York state license plate has three letters followed by four digits. How many different license plates are possible if the digits can’t be repeated and letters can be repeated? 17) The standard New York state license plate has three letters followed by four digits. How many different license plates are possible if neither the digits nor the letters can be repeated? 18) Students at a small college are assigned student identification numbers. The IDs are made up of five digits. If the digits can be repeated, how many identification numbers are possible? 19) You are assigned a computer generated 4-digit password to access your new voice mail account. If the digits can be repeated, how many passwords are possible? #20 – 30: Counting principle problems with sample space construction and probability questions. Section 4.3 Continued: 20) A couple plans to have three children. a) Determine the number of points in the sample space of the possible arrangements of boys and girls. b) Construct a tree diagram and list the sample space. Find the probability the couple has: c) d) e) Two girls At least one girl A girl then two boys 21) A couple plans to have two children. a) Determine the number of points in the sample space of the possible arrangements of boys and girls. b) Construct a tree diagram and list the sample space. Find the probability the couple has: c) Two girls d) At least one girl e) A girl then a boy 22) Answer each part of question 21, but assume the family plans to have 4 children. 23) A coin is tossed three times and a sequence of heads and tails is recorded. a) Determine the number of points in the sample space b) Construct a tree diagram and list the sample space Find the probability that c) d) e) No heads are tossed Exactly one head is tossed Three heads are tossed 24) Answer each part of question 23 but assume a coin is tossed two times. 25) Two six sided dice are rolled and the number on each face is recorded. a) Determine the number of points in the sample space b) c) d) e) f) Construct a tree diagram and list the sample space Find the probability that a double is rolled (both dice have the same number) Find the probability that a sum of 7 is rolled Find the probability a sum of 2 is rolled Are you as likely to roll a sum of 2 as you are of rolling a sum or 7 26) Answer each part of question 25 but assume the first dice is a 5 sided dice and the second dice is a four sided dice. 27) A coin is flipped then a 6 sided dice is rolled. a) Determine the number of points in the sample space b) Construct a tree diagram and list the sample space c) Find the probability that a head is flipped and an even number is rolled d) Find the probability that a tail is flipped and a number less than 3 is rolled 28) Answer question 27 but assume that a coin is flipped and a 5 sided dice is rolled. 29) A coin is flipped then a number is picked out of a hat containing the numbers 1,2 and 3. a) Determine the number of points in the sample space b) Construct a tree diagram and list the sample space c) Find the probability that a head is flipped and an even number is selected d) Find the probability that a tail is flipped and a number less than 3 is selected 30) Answer question 29 but assume that a coin is flipped and a number is picked out of a hat containing the numbers 1,2,3 and 4. Section 4.3 answers: 1a) 365*365 = 133,225 ways 3) 5*3*4 = 60 outfits 7) 3*3 = 9 11) 4*5*3=60 1b) 365*364 = 132860 ways 5a) 10*10 = 100 5b) 10*9 = 90 9) 26*26*10*10*10 = 676,000 13) 5*4 = 20 15) 26*26*10*10*10*10= 6,760,000 17) 26*25*24*10*9*8*7 = 78,624,000 19) 10*10*10*10=10,000 21a) 2*2 = 4 21b) {BB BG GB GG} 21c) ¼ 21d) ¾ 21e) ¼ 23a) 2*2*2 = 8 23a) {HHH HHT HTH HTT THH THT TTH TTT} 23c) 1/8 23d) 3/8 23e) 1/8 25a) 6*6 = 36 25b) (1,1) sum 2 (1,2) sum 3 (1,3) sum 4 (1,4) sum 5 (1,5) sum 6 (1,6) sum 7 (2,1) sum 3 (2,2) sum 4 (2,3) sum 5 (2,4) sum 6 (2,5) sum 7 (2,6) sum 8 (3,1) sum 4 (4,1) sum 5 (5,1) sum 6 (6,1) sum 7 (3,2) sum 5 (4,2) sum 6 (5,2) sum 7 (6,2) sum 8 (3,3) sum 6 (4,3) sum 7 (5,3) sum 8 (6,3) sum 9 25c) 6/36 = 1/6 25d) 6/36 = 1/6 (3,4) sum 7 (3,5) sum 8 (3,6) sum 9 (4,4) sum 8 (4,5) sum 9 (4,6) sum 10 (5,4) sum 9 (5,5) sum 10 (5,6) sum 11 (6,4) sum 10 (6,5) sum 11 (6,6) sum 12 25e) 1/36 27a) 2*6 = 12 27b) {H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6} 27c) 3/12 = ¼ 27d) 2/12 = 1/6 29a) 2*3 = 6 29b) {H1 H2 H3 T1 T2 T3} 29c) 1/6 29d) 2/6=1/3 25f) no 7 is more likely Section 4.4: “Or problems” #1-4: One card is selected at random from a standard deck of playing cards. Determine the probability of: P (A or B). Hint use the formula: P(A or B) = P(A) + P(B) – P(A and B) 1. A = an ace B = a jack 2. A = a red card 3. A = an ace 4. A = a picture card (J,Q,K) B = a diamond B = a black card B = a red card Homework #5-8 use the formula P(A or B) = P(A) + P(B) – P(A and B) to answer the following. 5) If P(A) = 0.5 and P(B) = 0.6 and P(A and B) = 0.2 find P(A or B) 6) If P(A) = 0.6 and P(B) = 0.8 and P(A and B) = 0.5 find P(A or B) 7) If P(A or B) = 0.8, P(A) = 0.5, P(B) = 0.6 find P(A and B) 8) If P(A or B) = 0.6 and P(B) = 0.3 and P(A and B) = 0.1 find P(A) Homework: #9 – 12: a single dice is rolled one time. Find the probability of rolling 9) An odd number or a number greater than 2 10) An even number or a number less than 5 11) A number less than 3 or greater than 5 12) A number greater than 5 or less than 4 Homework #13 – 18: One card is selected from a deck of playing cards. Find the probability of selecting 13) A jack or a diamond 14) A queen or a red card 15) A seven or a six 16) A queen or a two 17) A red card or a ten 18) A black card or a queen Homework #19 – 22: One marble is selected from a bag that contains 4 red, 5 white, 6 blue and 2 yellow. Find the probability of selecting 19) A red or a white 21) A blue or a white 20) A red or a blue 22) A yellow or a blue Section 4.4 Continued: Homework #23 – 26: The numbers 1, 2, 3, 4, …., 14 are placed on a piece of paper and thrown into a bucket. One piece of paper is randomly selected. What is the probability that: 23) The number is odd or greater then 7 24) The number is even or less than 8 25) The number is greater than or equal to 5 or even 26) The number is less than or equal to 8 or odd Homework #27-30: The letters of the word Mississippi are placed on pieces of paper and thrown into a bucket. One piece of paper is randomly selected. What is the probability that: 27) The letter is a p or an m? 28) The letter is a vowel or an m? 29) The letter is a vowel or alphabetically after r? 30) The letter is a vowel or alphabetically before r? Section 4.4 Answers: 1) 8/52 or 2/13 3) 16/52 = 4/13 5) 0.9 7) 0.3 9) 5/6 11) 3/6 = ½ 13) 16/52 = 4/13 15) 8/52 = 2/13 17) 28/52 = 7/13 19) 9/17 21) 11/17 23) 11/14 25) 12/14 = 6/7 27) 3/11 29) 8/11 Section 4.5 “AND problems” Homework # 1 – 10: Two cards are selected at random from a deck of 52 cards WITHOUT replacement. Find the requested probabilities, (write your answer as a reduced fraction.) 1) Both cards are red 2) Both cards are black 3) The first card is a 7 and the second card is a 4 4) The first card is a queen and the second card is a 3 5) Both cards are fives 6) Both cards are face cards 7) The first card is red and the second card is black 8) The first card is a heart and the second card is a spade 9) The first card is not a seven and the second card is a seven 10) The first card is not a four and the second card is a four Homework # 11 – 20: Two cards are selected at random from a deck of 52 cards WITH replacement. Find the requested probabilities (write your answer as a reduced fraction) 11) Both cards are hearts 12) Both cards are spades 13) The first card is a six and the second card is black 14) The first card is black and the second card is a six 15) The first card is a face card and the second card is not a face card 16) Both cards are face cards 17) Neither card is black 18) Neither card is a spade 19) The first card is a queen and the second card is a seven 20) The first card is not a queen and the second card is not a seven Section 4.4 Continued: Homework # 21-24: The spinner below is spun twice, find the requested probabilities (write your answer as a reduced fraction.) 21) Both spins are yellow 22) Both spins are purple 23) The first spin is green and the second is not red 24) The first spin is not yellow and the second spin is red Homework # 25-28: The spinner below is spun three times, find the requested probabilities (write your answer as a reduced fraction.) 25) The spins in order are green, yellow the blue 26) The spins in order are green, green and yellow 27) Green is spun each time 28) Yellow is spun each time Section 4.5 Continued: Homework #29 – 32: A coin is tossed then a dice is rolled, find the requested probabilities (write your answer as a reduced fraction.) 29) The coin is a head and the dice is a 4 30) The coin is a tail and the dice is not a 4 31) The coin is not a head and the dice is an even number 32) The coin is not a tail and the dice is greater than 2 Homework #33-34: A sample of 30 women who recently had a home built yielded the following information about their builder: Number of women Would you recommend builder to a friend 19 Yes 6 No 5 Not sure Three women who provided information for the table were selected at random. Find the probability that: 33) The first would not recommend her home builder, but the second and third would recommend their home builder. 34) The first would recommend her home builder, but the second and third would not recommend their home builder. Homework #35-38 Two marbles are drawn from a bag that contains 5 blue, 3 red and 2 purple marbles without replacement. Find the requested probabilities 35) Both are blue 36) Both are red 37) The first is purple and the second is not blue 38) The first is purple and the second is red Section 4.5 Continued: Homework #39-42 Two marbles are drawn from a bag that contains 5 blue, 3 red and 2 purple marbles with replacement. Find the requested probabilities 39) Both are blue 40) Both are red 41) The first is purple and the second is not blue 42) The first is not purple and the second is red Section 4.5 answers, with a bit of work: 1) 26 25 ∗ 52 51 25 = 102 3) 11) 13 52 21) 1 5 ∗ = 1 5 1 25 23) 29) 1 2 ∗ = 1 6 1 12 31) 39) 13 4 4 ∗ 52 51 1 4 52 ∗ 52 = 16 13) 5 5 ∗ 10 10 1 =4 1 5 1 2 41) 4 = 663 5) 26 4 3 ∗ 52 51 12 52 ∗ 52 = 169 17) 4 5 1 4 1 4 4 25 ∗ = 3 6 1 4 2 5 ∗ 10 10 25) 33) 1 = 10 1 2 40 26 26 ∗ 52 51 ∗ 52 = 26 15) ∗ = 1 1 = 221 7) ∗ ∗ = 6 30 ∗ 30 1 32 19 18 ∗ 29 28 = 27) 171 2030 1 2 13 = 51 9) 26 26 ∗ 52 52 1 48 52 = 4 19) 1 2 1 2 1 8 35) 5 10 ∗ = ∗ ∗ = 4 9 2 9 4 16 ∗ 51 = 221 4 52 37) 4 1 ∗ 52 = 169 2 10 4 9 ∗ = 4 45 Section 4.6: The Counting Principle and Permutations 1) The daily double at most racetracks consists of selecting the winning horse in both the first and the second race. If the first race has 7 entries and the second has 8 entries, how many daily double tickets mush you purchase to guarantee a win? 2) If a club consists of 10 members, how many different arrangements of president, vice-president and secretary are possible? 3) A lock combination consists of three numbers between 0 and 24 inclusive. How many combinations are possible, assume repetition is allowed? 4) A lock combination consists of three numbers between 0 and 24 inclusive. How many combinations are possible, assume repetition is NOT allowed? 5) A social security number consists of nine digits. How many different social security numbers are possible if repetition of digits is NOT permitted? 6) A social security number consists of nine digits. How many different social security numbers are possible if repetition of digits is permitted? 7) A class ring has three stone options (emerald, ruby or sapphire), two metal options (gold and silver). How many different class rings are possible? 8) Anthony has 6 dress shirts and 4 ties. How many shirt and tie outfits can he make? 9) How many different sandwiches can be made with 2 choices of bread, 3 choices of toppings and 2 choices of meat? 10) A batting order must be created from the 9 starters on a baseball team. How many batting orders are possible? 11) A batting order must be created from the 9 starters on a baseball team. How many batting orders are possible if the pitcher must bat last? 12) A batting order must be created from the 9 starters on a baseball team. How many batting orders are possible is a specific person must hit 4th? 13) A batting order must be created from the 9 starters on a baseball team. How many batting orders are possible if the first 4 batters are already assigned? 14) A multiple choice test has 5 questions and each question has 4 choices. How many ways are there to answer the questions? 15) A multiple choice test has 5 questions and each question has 4 choices. How many ways are there to answer the questions if you know the answer to the first question? Section 4.6 continued: 16) A multiple choice test has 5 questions and each question has 4 choices. How many ways are there to answer the questions if you know the answer to the first two questions? Homework 17 – 20: an identification code is to consist of two letters followed by 4 digits. How many different codes are possible if 17) Repetition is permitted 18) The first letter must be A, B, C or D and repetition is not permitted? 19) Repetition of numbers is permitted, but repetition of letters is not permitted? 20) Repetition is not permitted? Homework #21-24: a license plate is to consist of three digits followed by two uppercase letters. Determine the number of different license plates possible if 21) Repetition of numbers and letters is not permitted 22) Repetition of numbers and letters is permitted 23) The first digit cannot be a zero, and repetition is not permitted 24) The last digit must be even and repetition is permitted Homework #25-30: Use the permutation of distinct objects formula following 25) How many ways can the letters of Mississippi be arranged? 26) How many ways can the letters of Pennsylvania be arranged? 27) How many ways can the letters x,x,x,y,y,z,z,z,z be arranged? 28) How many ways can the letters a,b,b,b,c,c,c,d,d,e be arranged? 29) How many ways can the letters of abracadabra be arranged? 30) How many ways can the letters of Diamondbacks be arranged? n! to solve the n1 ! n2 ! n3 ! nr ! Section 4.6 Answers: 1) 7*8 = 56 3) 25*25*25= 15,625 5) 10*9*8*7*6*5*4*3*2 = 3,628,800 ways 7) 3*2 = 6 9) 2*3*2 = 12 11) 8*7*6*5*4*3*2*1*1 =40,320 13) 1*1*1*1*5*4*3*2*1 = 120 15) 1*4*4*4*4 = 256 17) 26*26*10*10*10*10 =6,760,000 19) 26*25*10*10*10*10 = 6,500,000 21) 10*9*8*26*25 = 468,000 23) 9*9*8*26*25 = 421,200 25) 27) 9! = 3!∗2!∗4! 1260 29) 11! = 5!∗2!∗2! 11! = 4!∗4!∗2! 83,160 34,650 Section 4.7: Conditional Probability Homework #1 – 8: Create a reduced sample space and find the requested probabilities. A family has three children. Assuming that boys and girls are equally likely, determine the probability that the family has… (Write your answer as a reduced fraction.) 1) Two girls given the first child is a girl 2) Two girls given the first child is a boy 3) Less than 2 girls given the first child is a girl 4) More than 2 girls given the first child is a girl. 5) Two boys given the middle child is a boy. 6) 1 boy given the middle child is a girl 7) 1 girl given the oldest child is a girl 8) Two girls given the oldest child is a girl. Homework #9 – 16: Create a reduced sample space and find the requested probabilities. A pair of dice is rolled and the sum of the dice is recorded, determine the probability that: (Write your answer as a reduced fraction.) 9) The sum is greater than 5 given the first dice is a 4. 10) The sum is greater than 9 given the second dice is a 6. 11) The sum is even given the second dice is a 4 12) The sum is odd given the first dice is a 3 13) A double is rolled given neither dice is a 4 14) A double is not rolled given the sum is greater than 10 15) The sum is less than 3 given the first dice is a 2 16) The sum is greater than 10 given the first dice is a 3 Section 4.7 Continued: Homework #17-24: Create a reduced sample space and find the requested probabilities. One card is selected from a deck of cards find the probability that: (Write your answer as a reduced fraction.) 17) The card is heart given it is not black 18) The card is a spade given it is red 19) The card is a four given that it is black 20) The card is a queen given that it is a face card 21) The card is not a seven given it is between 3 and 8 inclusive 22) The card is not a seven given it is between 3 and 8 exclusive 23) The card is not red given it is a queen 24) The card is not black given it is a two Homework: #25 – 28 Assume that a hat contains 4 bills: a $1 bill, a $5 bill, a $10 bill and a $20 bill. Two bills are to be selected at random with replacement. Construct a sample space, and find the probability that: (Write your answer as a reduced fraction.) 25) Both bills are $1 bills if given the first selected is a $1 bill 26) Both bills have a value greater than a $5 bill given the second bill is a $10 bill. 27) The second bill is a five given the first bill is a $5. 28) The first bill is a five given the second bill is a $5. Section 4.7 Continued: Homework #29 – 32: At a zoo, a sampling of children was asked if the zoo were to get one additional animal, would they prefer a lion or an elephant. The results of the survey follow: (Write your answer as a reduced fraction.) Lion Elephant Total Boys 90 110 200 Girls 75 85 160 Total 165 195 360 If one child who was in the survey is selected at random, find the probability that 29) The child selected the lion, given the child is a girl. 30) The child is a boy, given the child preferred the elephant. 31) The child selected is a girl, given that the child preferred the lion. 32) The child preferred the elephant given they are a girl. Homework #33-36: A quality control inspector, is checking a sample of light bulbs for defects. The following table summarizes her findings. (Write your answer as a reduced fraction.) Wattage Good Defective Total 20 80 15 95 50 100 5 105 100 120 10 130 Total 300 30 330 If one of these light bulbs is selected at random, find the probability that the light bulb is 33) Good given it is 100 watts 35) 100 watts given it is good 34) Defective given it is 50 watts 36) 50 watts given it is defective Section 4.7 Continued: Homework #37 – 40: A survey of students to determine if they had a pierced ear was given. The results are summarized in the table below. (Write your answer as a reduced fraction.) If one person is selected at random find the probability that 37) Female given they are pierced 38) Male given they are not pierced 39) Not pierced given they are female 40) Not pierced given they are male Section 4.7 Answers: 1) 2/4 = ½ 3) 1/4 5) 2/4 = ½ 7) ¼ 9) 5/6 11) 3/6 = ½ 13) 5/25 = 1/5 15) 0/6 = 0 17) 13/26 = ½ 19) 2/26 = 1/13 21) 20/24 = 5/6 23) 2/4 = ½ 25) ¼ 27) ¼ 29) 75/160 = 15/32 31) 75/165 5/11 33) 120/130 = 12/13 35) 120/300 = 2/5 37) 288/324 = 8/9 39) 32/320 = 1/10