Chapter 14

Monkeys Typing Shakespeare An often-quoted example of an event of extremely low probability is that a monkey types Shakespeare's entire play Hamlet by randomly striking keys on a typewriter. Assume that the typewriter has 48 keys (including the space bar) and that the monkey is equally likely to hit any key. (a) Find the probability that such a monkey will actually correctly type just the title of the play as his first word. (b) What is the probability that the monkey will type the phrase "To be or not to be" as his first words?

Five-Letter Words Five-letter "words" are formed using the letters \(A, B, C, D, E, F, G .\) How many such words are possible for each of the following conditions? (a) No condition is imposed. (b) No letter can be repeated in a word. (c) Each word must begin with the letter \(A\) . (d) The letter \(C\) must be in the middle. (e) The middle letter must be a vowel.

Hitting a Target An archer normally hits the target with probability of \(0.6 .\) She hires a new coach for a series of special lessons. After the lessons she hits the target in five out of eight attempts. (a) Find the probability that she would have hit five or more out of the eight attempts before her lessons with the new coach. (b) Did the new coaching effective if the probability in part (a) is 0.05 or less.)

Making Words A monkey is trained to arrange wooden blocks in a straight line. He is then given six blocks showing the letters \(A, E, H, L, M, T\) . What is the probability that he will arrange them to spell the word HAMLET?

Palindromes How many five-letter palindromes are possible? (A palindrome is a string of letters that reads the same backward and forward, such as the string \(X C Z C X\) .)

These problems involve distinguishable permutations. Arrangements In how many ways can two blue marbles and four red marbles be arranged in a row?

Making Words A monkey is trained to arrange wooden blocks in a straight line. She is then given 11 blocks showing the letters \(A, B, B, I, I, L, O, P, R, T, Y .\) What is the probability that the monkey will arrange the blocks to spell the word PROBABILITY?

Names of Variables A certain computer programming language allows names of variables to consist of two characters, the first being any letter and the second being any letter or digit. How many names of variables are possible?

These problems involve distinguishable permutations. Arrangements In how many different ways can five red balls, two white balls, and seven blue balls be arranged in a row?

Horse Race Eight horses are entered in a race. You randomly predict a particular order for the horses to complete the race. What is the probability that your prediction is correct?

These problems involve distinguishable permutations. Arranging Coins In how many different ways can four pennies, three nickels, two dimes, and three quarters be arranged in a row?

Genetics Many genetic traits are controlled by two genes, one dominant and one recessive. In Gregor Mendel's original experiments with peas, the genes controlling the height of the plant are denoted by T (tall) and t (short). The gene T is dominant, so a plant with the genotype (genetic makeup) TT or Tt is tall, whereas one with genotype tt is short. By a statistical analysis of the offspring in his experiments, Mendel concluded that offspring inherit one gene from each parent and that each possible combination of the two genes is equally likely. If each parent has the genotype Tt, then the following chart gives the possible genotypes of the offspring: Find the probability that a given offspring of these parents will be (a) tall or (b) short.

Seating Arrangements In how many ways can four men and four women be seated in a row of eight seats for the following situations? (a) The women are to be seated together, and the men are to be seated together. (b) They are to be seated alternately by gender.

These problems involve distinguishable permutations. Arranging Letters In how many different ways can the letters of the word ELEEMOSYNARY be arranged?

Arranging Books In how many ways can five different mathematics books be placed on a shelf if the two algebra books are to be placed next to each other?

These problems involve distinguishable permutations. Distributions A man bought three vanilla ice-cream cones, two chocolate cones, four strawberry cones, and five butterscotch cones for his 14 children. In how many ways can he distribute the cones among his children?

\(41-42\) . Determine whether the events \(E\) and \(F\) in the given experiment are mutually exclusive. The experiment consists of selecting a person at random. (a) \(E :\) The person is male. F: The person is female. (b) \(E :\) The person is tall. \(F :\) The person is blond.

Arranging Books Eight mathematics books and three chemistry books are to be placed on a shelf. In how many ways can this be done if the mathematics books are next to each other and the chemistry books are next to each other?

\(41-42\) . Determine whether the events \(E\) and \(F\) in the given experiment are mutually exclusive. The experiment consists of choosing at random a student from your class. (a) \(E :\) The student is female. F: The student wears glasses. (b) \(E :\) The student has long hair. \(F :\) The student is male.

Three-Digit Numbers Three-digit numbers are formed using the digits \(2,4,5,\) and \(7,\) with repetition of digits allowed. How many such numbers can be formed if (a) the numbers are less than 700\(?\) (b) the numbers are even? (c) the numbers are divisible by 5\(?\)

These problems involve distinguishable permutations. Work Assignments Eight workers are cleaning a large house. Five are needed to clean windows, two to clean the carpets, and one to clean the rest of the house. In how many different ways can these tasks be assigned to the eight workers?

\(43-44\) . A die is rolled, and the number showing is observed. Determine whether the events \(E\) and \(F\) are mutually exclusive. Then find the probability of the event \(E \cup F .\) (a) \(E :\) The number is even. F: The number is odd. (b) \(E :\) The number is even. F: The number is greater than \(4 .\)

Three-Digit Numbers How many three-digit odd numbers can be formed using the digits \(1,2,4,\) and 6 if repetition of digits is not allowed?

These problems involve distinguishable permutations. Jogging Routes A jogger jogs every morning to his health club, which is eight blocks east and five blocks north of his home. He always takes a route that is as short as possible, but he likes to vary it (see the figure). How many different routes can he take? [Hint: The route shown can be thought of as ENNEEENENEENE, where \(E\) is East and \(N\) is North.]

\(43-44\) . A die is rolled, and the number showing is observed. Determine whether the events \(E\) and \(F\) are mutually exclusive. Then find the probability of the event \(E \cup F .\) (a) \(E :\) The number is greater than \(3 .\) \(F :\) The number is less than 5 . (b) \(E :\) The number is divisible by \(3 .\) F: The number is less than \(3 .\)

Pairs of Initials Explain why in any group of 677 people, at least two people must have the same pair of initials.

These problems involve combinations. Choosing Books In how many ways can three books be chosen from a group of six different books?

\(45-46\) A card is drawn at random from a standard 52 -card deck. Determine whether the events \(E\) and \(F\) are mutually exclusive. Then find the probability of the event \(E \cup F .\) (a) \(E :\) The card is a face card. F: The card is a spade. (b) \(E :\) The card is a heart. F: The card is a spade.

These problems involve combinations. Pizza Toppings In how many ways can three pizza toppings be chosen from 12 available toppings?

\(45-46\) A card is drawn at random from a standard 52 -card deck. Determine whether the events \(E\) and \(F\) are mutually exclusive. Then find the probability of the event \(E \cup F .\) (a) \(E :\) The card is a club. F: The card is a king. (b) \(E :\) The card is an ace. F: The card is a spade.

These problems involve combinations. Committee In how many ways can a committee of three members be chosen from a club of 25 members?

These problems involve combinations. Choosing a Group In how many ways can six people be chosen from a group of ten?

These problems involve combinations. Draw Poker Hands How many different five- card hands can be dealt from a deck of 52 cards?

Roulette An American roulette wheel has 38 slots. Two of the slots are numbered 0 and \(00,\) and the rest are numbered from 1 to \(36 .\) Find the probability that the ball lands in an odd-numbered slot or in a slot with a number higher than \(31 .\)

These problems involve combinations. Stud Poker Hands How many different seven-card hands can be picked from a deck of 52 cards?

Making Words \(A\) toddler has eight wooden blocks showing the letters \(A, E, I, G, L, N, T,\) and \(R .\) What is the probability that the child will arrange the letters to spell one of the words \(T R I A N G L E\) or \(I N T E G R A L ?\)

These problems involve combinations. Choosing Exam Questions A student must answer seven of the ten questions on an exam. In how many ways can she choose the seven questions?

Choosing a Committee A committee of five is chosen randomly from a group of six males and eight females. What is the probability that the committee includes either all males or all females?

These problems involve combinations. Three-Topping Pizzas A pizza parlor offers a choice of 16 different toppings. How many three-topping pizzas are possible?

These problems involve combinations. Violin Recital A violinist has practiced 12 pieces. In how many ways can he choose eight of these pieces for a recital?

Marbles in a Jar A jar contains six red marbles numbered 1 to 6 and ten blue marbles numbered 1 to \(10 .\) A marble is drawn at random from the jar. Find the probability that the given event occurs. (a) The marble is red. (b) The marble is odd-numbered. (c) The marble is red or odd-numbered. (d) The marble is blue or even-numbered.

These problems involve combinations. Choosing Clothing If a woman has eight skirts, in how many ways can she choose five of these to take on a weekend trip?

A coin is tossed twice. Let \(E\) and \(F\) be the following events: $$\begin{array}{l}{\text { E. The first toss shows heads. }} \\ {\text { F: The second toss shows heads. }}\end{array}$$ (a) Are the events \(E\) and \(F\) independent? (b) Find the probability of showing heads on both tosses.

These problems involve combinations. Field Trip In how many ways can seven students from a class of 30 be chosen for a field trip?

A die is rolled twice. Let \(E\) and \(F\) be the following events: $$\begin{array}{l}{\text { E. The first roll shows a six. }} \\ {\text { F. The second roll shows a six. }}\end{array}$$ (a) Are the events \(E\) and \(F\) independent? (b) Find the probability of showing a six on both rolls.

These problems involve combinations. Lottery In the 6\(/ 49\) lottery game, a player picks six numbers from 1 to \(49 .\) How many different choices does the player have?

A die is rolled twice. What is the probability of showing a one on both rolls?

Subsets A set has eight elements. (a) How many subsets containing five elements does this set have? (b) How many subsets does this set have?

A die is rolled twice. What is the probability of showing a one on the first roll and an even number on the second roll?

Travel Brochures A travel agency has limited numbers of eight different free brochures about Australia. The agent tells you to take any that you like but no more than one of any kind. In how many different ways can you choose brochures (including not choosing any)?