In Exercises 536, express all probabilities as fractions. Scheduling Routes A

Notation What does the symbol ! represent? Four different people can stand in a line 4! different ways. What is the actual number of ways that four people can stand in a line?

California Fantasy The winning numbers for the current California Fantasy 5 lottery are 13, 18, 22, 24, and 32 in any order. Do calculations for winning this lottery involve permutations or combinations? Why?

. California Daily 4 The winning numbers for the current California Daily 4 lottery are 5, 0, 0, and 4 in that exact order. Because order counts, do calculations for this lottery involve either of the two permutation rules presented in this section? Why or why not? If not, what rule does apply?

. Selections with Replacement When randomly selecting items, if successive selections are made with replacement of previously selected items, which of the five rules of this section apply: (1) fundamental couting rule (2) factorial rule (3) permutations rule (when all items are different) (4) permutations rule (when some items are identical to others) (5) combinations rule?

In Exercises 536, express all probabilities as fractions.ATM Pin Numbers A thief steals an ATM card and must randomly guess the correct pin code that consists of four digits (0 through 9) that must be entered in the correct order. Repetition of digits is allowed. What is the probability of a correct guess on the first try?

In Exercises 536, express all probabilities as fractions.Social Security Numbers A Social Security number consists of nine digits in a particular order, and repetition of digits is allowed. If randomly selecting digits for one Social Security number, what is the probability that you get the Social Security number of the president?

In Exercises 536, express all probabilities as fractions.Baseball Batting Order If you know the names of the starting batters for a baseball team, what is the probability of randomly selecting a batting order and getting the order that is used in the beginning of the game?

In Exercises 536, express all probabilities as fractions.Harry Potter Books There are seven books in the Harry Potter series. If the books are read in a randomly selected order, what is the probability that they are read in the order that they were written?

In Exercises 536, express all probabilities as fractions. Lady Antebellum Songs A fan of Lady Antebellum music plans to make a custom CD with 12 of her 27 songs. How many different combinations of 12 songs are possible? Is it practical to make a different CD for each possible combination?

In Exercises 536, express all probabilities as fractions.Blackjack In the game of blackjack played with one deck, a player is initially dealt 2 different cards from the 52 different cards in the deck. Find the probability of getting a 2card initial hand consisting of the ace of clubs and the ace of spades in any order.

In Exercises 536, express all probabilities as fractions. Scheduling Routes A presidential candidate plans to begin her campaign by visiting the capitals in 4 of the 50 states. What is the probability that she selects the route of Saramento, Albany, Juneau, and Hartford? Is it practical to list all of the different possible routes in order to select the one that is best?

In Exercises 536, express all probabilities as fractions. FedEx Deliveries With a short time remaining in the day, a FedEx driver has time to make deliveries at three locations among the eight locations remaining. How many different routes are possible?

In Exercises 536, express all probabilities as fractions.Classic Counting Problem A classic counting problem is to determine the number of different ways that the letters of Mississippi can be arranged. Find that number.

In Exercises 536, express all probabilities as fractions.Statistics Count How many different ways can the letters of statistics be arranged?Statistics Count How many different ways can the letters of statistics be arranged?

In Exercises 536, express all probabilities as fractions.Connecticut Lottery Winning the jackpot in the Connecticut Classic Lotto requires that you choose six different numbers from 1 to 44, and your numbers must match the same six numbers that are later drawn. The order of the selected numbers does not matter. If you buy one ticket, what is the probability of winning the jackpot?

In Exercises 536, express all probabilities as fractions.Florida Lottery Winning the jackpot in the Florida Lotto requires that you choose six different numbers from 1 to 53, and your numbers must match the same six numbers that are later drawn. The order of the selected numbers does not matter. If you buy one ticket, what is the probability of winning the jackpot?

In Exercises 536, express all probabilities as fractions.Teed Off When four golfers are about to begin a game, they often toss a tee to randomly select the order in which they tee off. What is the probability that they tee off in alphabetical order?

In Exercises 536, express all probabilities as fractions.Stacking Books The author currently has seven different books in print. If those seven books are stacked in a random order, what is the probability that they are arranged in alphabetical order from top to bottom?

In Exercises 536, express all probabilities as fractions. Maine Lottery a. In the Maine Megabucks game, you win the jackpot by selecting five different whole numbers from 1 through 41 and getting the same five numbers (in any order) that are later drawn. What is the probability of winning a jackpot in this game? b. In the Maine Pick 4 game, you win a straight bet by selecting four digits (with repetition allowed) and getting the same four digits in the exact same order they are later drawn. What is the probability of winning this game? c. The Maine Pick 4 game returns $5000 for a winning $1 ticket. What should be the return if Maine were to run this game for no profit?

In Exercises 536, express all probabilities as fractions. Illinois Lottery a. In the Illinois Little Lotto game, you win the jackpot by selecting five different whole numbers from 1 through 39 and getting the same five numbers (in any order) that are later drawn. What is the probability of winning a jackpot in this game? b. In the Illinois Pick 3 game, you win a bet by selecting three digits (with repetition allowed) and getting the same three digits in the exact same order as they are later drawn. What is the probability of winning this game? c. The Illinois Pick 3 game returns $500 for a winning $1 ticket. What should be the return if Illinois were to run this game for no profit?

In Exercises 536, express all probabilities as fractions.. Corporate Officials and Committees The Teknomill Corporation must appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO). It must also appoint a Planning Committee with four different members. There are 12 qualified candidates, and officers can also serve on the committee. a. How many different ways can the officers be appointed? b. How many different ways can the committee be appointed? c. What is the probability of randomly selecting the committee members and getting the 4 youngest of the qualified candidates?

In Exercises 536, express all probabilities as fractions.Phase I of a Clinical Trial A clinical test on humans of a new drug is normally done in three phases. Phase I is conducted with a relatively small number of healthy volunteers.For example, a phase I test of bexarotene involved only 14 subjects. Assume that we want to treat 14 healthy humans with this new drug and we have 16 suitable volunteers available. a. If the subjects are selected and treated in sequence, so that the trial is discontinued if anyone displays adverse effects, how many different sequential arrangements are possible if 14 people are selected from the 16 that are available? b. If 14 subjects are selected from the 16 that are available, and the 14 selected subjects are all treated at the same time, how many different treatment groups are possible? c. If 14 subjects are randomly selected and treated at the same time, what is the probability of selecting the 14 youngest subjects?

In Exercises 536, express all probabilities as fractions. Combination Lock The typical combination lock uses three numbers between 0 and 49, and they must be selected in the correct sequence. How many different combinations are possible? Which of the five rules of this section is used to find that number? Is the name of combination lock appropriate? If not, what other name would be better?

In Exercises 536, express all probabilities as fractions.Safety with Numbers The author owns a safe in which he stores all of his great ideas for the next edition of this book. The safe combination consists of four numbers between 0 and 99, and the safe is designed so that numbers can be repeated. If another author breaks in and tries to steal these ideas, what is the probability that he or she will get the correct combination on the first attempt? Assume that the numbers are randomly selected. Given the number of possibilities, does it seem feasible to try opening the safe by making random guesses for the combination?

In Exercises 536, express all probabilities as fractions.Jumble Puzzle Many newspapers carry Jumble, a puzzle in which the reader must unscramble letters to form words. The letters MYAIT were included in newspapers on the day this exercise was written. How many ways can those letters be arranged? Identify the correct unscrambling then determine the probability of getting that result by randomly selecting one arrangement of the given letters.

In Exercises 536, express all probabilities as fractions. Jumble Puzzle Repeat the preceding exercise using these letters: RAWHOR.

In Exercises 536, express all probabilities as fractions.Counting with Fingers How many different ways can you touch two or more fingers to each other on one hand?

In Exercises 536, express all probabilities as fractions. Identity Theft with Credit Cards Credit card numbers typically have 16 digits, but not all of them are random. Answer the following and express probabilities as fractions. a. What is the probability of randomly generating 16 digits and getting your MasterCard number? b. Receipts often show the last four digits of a credit card number. If those last four digits are known, what is the probability of randomly generating the other digits of your MasterCard number? c. Discover cards begin with the digits 6011. If you also know the last four digits of a Discover card, what is the probability of randomly generating the other digits and getting all of them correct? Is this something to worry about?

In Exercises 536, express all probabilities as fractions. Electricity When testing for current in a cable with five colorcoded wires, the author used a meter to test two wires at a time. How many different tests are required for every possible pairing of two wires?

In Exercises 536, express all probabilities as fractions.ATM You want to obtain cash by using an ATM machine, but it's dark and you can't see your card when you insert it. The card must be inserted with the front side up and the printing configured so that the beginning of your name enters first. a. What is the probability of selecting a random position and inserting the card with the result that the card is inserted correctly? b. What is the probability of randomly selecting the card's position and finding that it is incorrectly inserted on the first attempt, but it is correctly inserted on the second attempt? c. How many random selections are required to be absolutely sure that the card works because it is inserted correctly?

In Exercises 536, express all probabilities as fractions.DNA Nucleotides DNA (deoxyribonucleic acid) is made of nucleotides. Each nucleotide can contain any one of these nitrogenous bases: A (adenine), G (guanine), C (cytosine), T (thymine). If one of those four bases (A, G, C, T) must be selected three times to form a linear triplet, how many different triplets are possible? All four bases can be selected for each of the three components of the triplet

In Exercises 536, express all probabilities as fractions.World Cup Soccer Tournament Every four years, 32 soccer teams compete in a world tournament. a. How many games are required to get one championship team from the field of 32 teams? b. If you make random guesses for each game of the tournament, find the probability of picking the winner in every game.

In Exercises 536, express all probabilities as fractions.Powerball As of this writing, the Powerball lottery is run in 42 states. Winning the jackpot requires that you select the correct five numbers between 1 and 59 and, in a separate drawing, you must also select the correct single number between 1 and 39. Find the probability of winning the jackpot.

In Exercises 536, express all probabilities as fractions.Mega Millions As of this writing, the Mega Millions lottery is run in 42 states. Winning the jackpot requires that you select the correct five numbers between 1 and 56 and, in a separate drawing, you must also select the correct single number between 1 and 46. Find the probability of winning the jackpot.

In Exercises 536, express all probabilities as fractions.Designing Experiment Clinical trials of Nasonex involved a group given placebos and another group given treatments of Nasonex. Assume that a preliminary phase I trial is to be conducted with 10 subjects, including 5 men and 5 women. If 5 of the 10 subjects are randomly selected for the treatment group, find the probability of getting 5 subjects of the same sex. Would there be a problem with having members of the treatment group all of the same sex?

In Exercises 536, express all probabilities as fractions.Area Codes USA Today reporter Paul Wiseman described the old rules for the threedigit telephone area codes by writing about possible area codes with 1 or 0 in the second digit. (Excluded: codes ending in 00 or 11, for tollfree calls, emergency services, and other special uses.) Codes beginning with 0 or 1 should also be excluded. How many different area codes were possible under these old rules?

Computer Variable Names A common computer programming rule is that names of variables must be between one and eight characters long. The first character can be any of the 26 letters, while successive characters can be any of the 26 letters or any of the 10 digits. For example, allowable variable names are A, BBB, and M3477K. How many different variable names are possible?

Handshakes a. Five managers gather for a meeting. If each manager shakes hands with each other manager exactly once, what is the total number of handshakes? b. If n managers shake hands with each other exactly once, what is the total number of handshakes? c. How many different ways can five managers be seated at a round table? (Assume that if everyone moves to the right, the seating arrangement is the same.) d. How many different ways can n managers be seated at a round table?

Change for a Quarter How many different ways can you make change for a quarter? (Different arrangements of the same coins are not counted separately.)

Oldie but Goodie A secretary types 10 different letters and addresses 10 corresponding envelopes. If he is in a hurry and randomly inserts the letters into the envelopes, what is the probability that exactly 9 of the letters are in the correct envelopes? (Based on Mathematics Magazine, Vol. 23, No. 4, 1950)