Chapter 14

\(19-22\) . A poker hand, consisting of five cards, is dealt from a standard deck of 52 cards. Find the probability that the hand contains the cards described. An ace, king, queen, jack, and ten of the same suit (royal flush)

Pitchers and Catchers An all-star baseball team has a roster of seven pitchers and three catchers. How many pitcher-catcher pairs can the manager select from this roster?

These problems involve permutations. Class Officers In how many different ways can a president, vice president, and secretary be chosen from a class of 15 students?

Genders of Children The ratio of male to female births is in fact not exactly one-to-one. The probability that a newborn turns out to be a male is about \(0.52 .\) A family has ten children. (a) What is the probability that all ten children are boys? (b) What is the probability all are girls? (c) What is the probability that five are girls and five are boys?

Two balls are picked at random from a jar that contains three red and five white balls. Find the probability of the following events. (a) Both balls are red. (b) Both balls are white.

License Plates Standard automobile license plates in California display a nonzero digit, followed by three letters, followed by three digits. How many different standard plates are possible in this system?

These problems involve permutations. Contest Prizes In how many different ways can first, second, and third prizes be awarded in a game with eight contestants?

Education Level In a certain county 20\(\%\) of the population have a college degree. A jury consisting of 12 people is selected at random from this county. (a) What is the probability that exactly two of the jurors have a college degree? (b) What is the probability that three or more of the jurors have a college degree?

Three CDs are picked at random from a collection of 12 \(\mathrm{CDs}\) of which four are defective. Find the probability of the following events. (a) All three CDs are defective. (b) All three CDs are functivening properly.

Combination Lock A combination lock has 60 different positions. To open the lock, the dial is turned to a certain number in the clockwise direction, then to a number in the counterclockwise direction, and finally to a third number in the clockwise direction. If successive numbers in the combination cannot be the same, how many different combinations are possible?

These problems involve permutations. Seating Arrangements In how many different ways can six of ten people be seated in a row of six chairs?

Defective Light Bulbs The DimBulb Lighting Company manufactures light bulbs for appliances such as ovens and refrigerators. Typically, 0.5\(\%\) of their bulbs are defective. From a crate with 100 bulbs, three are tested. Find the probability that the given event occurs. (a) All three bulbs are defective. (b) One or more bulbs is defective.

A five-card poker hand is drawn from a standard 52 -card deck. Find the probability that at least one card is a spade.

True-False Test \(A\) true-false test contains ten questions. In how many different ways can this test be completed?

These problems involve permutations. Seating Arrangements In how many different ways can six people be seated in a row of six chairs?

Quality Control An assembly line that manufactures fuses for automotive use is checked every hour to ensure the quality of the finished product. Ten fuses are selected randomly, and if any one of the ten is found to be defective, the process is halted and the machines are recalibrated. Suppose that at a certain time 5\(\%\) of the fuses being produced are actually defective. What is the probability that the assembly line is halted at that hour's quality check?

A five-card poker hand is drawn from a standard 52-card deck. Find the probability that at least one card is a face card.

Ordering a Car An automobile dealer offers five models. Each model comes in a choice of four colors, three types of stereo equipment, with or without air conditioning, and with or without a sunroof. In how many different ways can a customer order an auto from this dealer?

These problems involve permutations. Three-Letter Words How many three-letter "words" can be made from the letters \(F G H I J K ?\) (Letters may not be repeated.)

Sick Leave The probability that a given worker at Dyno Nutrition will call in sick on a Monday is \(0.04 .\) The packaging department has eight workers. What is the probability that two or more packaging workers will call in sick next Monday?

Four Siblings A couple intends to have four children. Assume that having a boy and having a girl are equally likely events. (a) List the sample space of this experiment. (b) Find the probability that the couple has only boys. (c) Find the probability that the couple has only boys. (d) Find the probability that the couple has two boys and two girls. the same sex. (e) Find the probability that the couple has at least two girls.

Classifications The registrar at a certain university classifies students according to a major, minor, year \((1,2,3,4),\) and sex \((\mathrm{M}, \mathrm{F}) .\) Each student must choose one major and either one or no minor from the 32 fields taught at this university. How many different student classifications are possible?

These problems involve permutations. Letter Permutations How many permutations are possible from the letters of the word \(L O V E ?\)

Political Surveys In a certain county, 60\(\%\) of the voters are in favor of an upcoming school bond initiative. If five voters are interviewed at random, what is the probability that exactly three of them will favor the initiative?

These problems involve permutations. Three-Digit Numbers How many different three-digit whole numbers can be formed by using the digits \(1,3,5,\) and 7 if no repetition of digits is allowed?

Pharmaceuticals \(A\) drug that is used to prevent motion sickness is found to be effective about 75\(\%\) of the time. Six friends, prone to seasickness, go on a sailing cruise, and all take the drug. Find the probability of each event. (a) None of the friends gets seasick. (b) All of the friends get seasick. (c) Exactly three get seasick. (d) At least two get seasick.

Roulette An American roulette wheel has 38 slots; two slots are numbered 0 and \(00,\) and the remaining slots are numbered from 1 to \(36 .\) Find the probability that the ball lands in an odd-numbered slot.

License Plates A state has registered 8 million automobiles. To simplify the license plate system, a state employee suggests that each plate display only two letters followed by three digits. Will this system create enough different license plates for all the vehicles that are registered?

These problems involve permutations. Piano Recital A pianist plans to play eight pieces at a recital. In how many ways can she arrange these pieces in the program?

Reliability of a Machine A machine that is used in a manufacturing process has four separate components, each of which has a 0.01 probability of failing on any given day. If any component fails, the entire machine breaks down. Find the probability that on a given day the indicated event occurs. (a) The machine breaks down. (b) The machine does not break down. (c) Only one component does not fail.

Making Words \(A\) toddler has wooden blocks showing the letters \(C, E, F, H, N,\) and \(R\) . Find the probability that the child arranges the letters in the indicated order. (a) In the order FRENCH (b) In alphabetical order

License Plates A state license plate design has six places. Each plate begins with a fixed number of letters, and the remaining places are filled with digits. (For example, one letter followed by five digits, two letters followed by four digits, and so on.) The state has 17 million registered vehicles. (a) The state decides to change to a system consisting of one letter followed by five digits. Will this design allow for enough different plates to accommodate all the vehicles that are registered? (b) Find a system that will be sufficient if the smallest possible number of letters is to be used.

These problems involve permutations. Running a Race In how many different ways can a race with nine runners be completed, assuming that there is no tie?

Genetics Huntington's disease is a hereditary ailment caused by a recessive gene. If both parents carry the gene but do not have the disease, there is a 0.25 probability that an off- spring will fall victim to the condition. A newly wed couple find through genetic testing that they both carry the gene (but do not have the disease). If they intend to have four children, find the probability of each event. (a) At least one child gets the disease. (b) At least three of the children get the disease.

Lottery In the 6\(/ 49\) lottery game, a player selects six numbers from 1 to \(49 .\) What is the probability of picking the six winning numbers?

Class Executive In how many ways can a president, vice president, and secretary be chosen from a class of 30 students?

These problems involve permutations. Signal Flags A ship carries five signal flags of different colors. How many different signals can be sent by hoisting exactly three of the five flags on the ship's flagpole in different orders?

Selecting Cards Three cards are randomly selected from a standard 52 -card deck, one at a time, with each card replaced in the deck before the next one is picked. Find the probability of each event. (a) All three cards are hearts. (b) Exactly two of the cards are spades. (c) None of the cards is a diamond. (d) At least one of the cards is a club.

An Unlikely Event The president of a large company selects six employees to receive a special bonus. He claims that the six employees are chosen randomly from among the 30 employees, of whom 19 are women and 11 are men. What is the probability that no woman is chosen?

Class Executive In how many ways can a president, vice president, and secretary be chosen from a class of 20 females and 30 males if the president must be a female and the vice president must be a male?

These problems involve permutations. Contest Prizes In how many ways can first, second, and third prizes be awarded in a contest with 1000 contestants?

Smokers and Nonsmokers The participants at a mathematics conference are housed dormitory-style, five to a room. Because of an oversight, conference organizers forgot to ask whether the participants are smokers. In fact, it turns out that 30\(\%\) are smokers. Find the probability that Fred, a nonsmoking conference participant, will be housed with (a) Exactly one smoker (b) One or more smokers

Guessing on a Test An exam has ten true-false questions. A student who has not studied answers all ten questions by just guessing. Find the probability the student correctly answers the given number of questions. (a) All ten questions (b) Exactly seven questions

Committee Officers A senate subcommittee consists often Democrats and seven Republicans. In how many ways can a chairman, vice chairman, and secretary be chosen if the chairman must be a Democrat and the vice chairman must be a Republican?

These problems involve permutations. Class Officers In how many ways can a president, vice president, secretary, and treasurer be chosen from a class of 30 students?

Telephone Marketing A mortgage company advertises its rates by making unsolicited telephone calls to random numbers. About 2\(\%\) of the calls reach consumers who are interested in the company's services. A telephone consultant can make 100 calls per evening shift. (a) What is the probability that two or more calls will reach an interested party in one shift? (b) How many calls does a consultant need to make to ensure at least a 0.5 probability of reaching one or more interested parties? IHint: Use trial and error.

Quality Control To control the quality of their product, the Bright-Light Company inspects three light bulbs out of each batch of ten bulbs manufactured. If a defective bulb is found, the batch is discarded. Suppose a batch contains two defective bulbs. What is the probability that the batch will be discarded?

Social Security Numbers Social Security numbers consist of nine digits, with the first digit between 0 and \(6,\) inclusive. How many Social Security numbers are possible?

These problems involve permutations. Seating Arrangements In how many ways can five students be seated in a row of five chairs if Jack insists on sitting in the first chair?

Effectiveness of a Drug \(A\) certain disease has a mortality rate of 60\(\%\) . A new drug is tested for its effectiveness against this disease. Ten patients are given the drug, and eight of them recover. (a) Find the probability that eight or more of the patients would have recovered without the drug. (b) Does the drug appear to be effective? (Consider the drug effective if the probability in part (a) is 0.05 or less.)