Chapter 11

In Exercises 1-2, the numbers that each pointer can land on and their respective probabilities are shown. Compute the expected value for the number on which each pointer lands. $$ \begin{array}{|c|c|} \hline \text { Outcome } & \text { Probability } \\ \hline 1 & \frac{1}{2} \\ \hline 2 & \frac{1}{4} \\ \hline 3 & \frac{1}{4} \\ \hline \end{array} $$

In Exercises 1-6, you are dealt one card from a 52-card deck. Find the probability that you are not dealt an ace.

Martha, Lee, Nancy, Paul, and Armando have all been invited to a dinner party. They arrive randomly, and each person arrives at a different time. a. In how many ways can they arrive? b. In how many ways can Martha arrive first and Armando last? c. Find the probability that Martha will arrive first and Armando last.

In Exercises 1-4, does the problem involve permutations or combinations? Explain your answer. (It is not necessary to solve the problem.) A medical researcher needs 6 people to test the effectiveness of an experimental drug. If 13 people have volunteered for the test, in how many ways can 6 people be selected?

Use the Fundamental Counting Principle to solve Exercises 1-12. Six performers are to present their comedy acts on a weekend evening at a comedy club. How many different ways are there to schedule their appearances?

A restaurant offers eight appetizers and ten main courses. In how many ways can a person order a two-course meal?

The numbers that each pointer can land on and their respective probabilities are shown. Compute the expected value for the number on which each pointer lands. $$ \begin{array}{|c|c|} \hline \text { Outcome } & \text { Probability } \\ \hline 1 & \frac{1}{8} \\ \hline 2 & \frac{1}{8} \\ \hline 3 & \frac{1}{2} \\ \hline 4 & \frac{1}{4} \\ \hline \end{array} $$

You are dealt one card from a 52-card deck. Find the probability that you are not dealt a \(3 .\)

Three men and three women line up at a checkout counter in a store. a. In how many ways can they line up? b. In how many ways can they line up if the first person in line is a woman, and then the line alternates by genderthat is a woman, a man, a woman, a man, and so on? c. Find the probability that the first person in line is a woman and the line alternates by gender.

Does the problem involve permutations or combinations? Explain your answer. (It is not necessary to solve the problem.) Fifty people purchase raffle tickets. Three winning tickets are selected at random. If first prize is \(\$ 1000\), second prize is \(\$ 500\), and third prize is \(\$ 100\), in how many different ways can the prizes be awarded?

Use the Fundamental Counting Principle to solve Exercises 1-12. Five singers are to perform on a weekend evening at a night club. How many different ways are there to schedule their appearances?

The model of the car you are thinking of buying is available in nine different colors and three different styles (hatchback, sedan, or station wagon). In how many ways can you order the car?

The tables in Exercises 3-4 show claims and their probabilities for an insurance company. a. Calculate the expected value and describe what this means in practical terms. b. How much should the company charge as an average premium so that it breaks even on its claim costs? c. How much should the company charge to make a profit of \(\$ 50\) per policy? PROBABILITIES FOR HOMEOWNERS' INSURANCE CLAIMS $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Amount of Claim (to the } \\ \text { nearest } \$ 50,000) \end{array} & \text { Probability } \\ \hline \$ 0 & 0.65 \\ \hline \$ 50,000 & 0.20 \\ \hline \$ 100,000 & 0.10 \\ \hline \$ 150,000 & 0.03 \\ \hline \$ 200,000 & 0.01 \\ \hline \$ 250,000 & 0.01 \\ \hline \end{array} $$

You are dealt one card from a 52-card deck. Find the probability that you are not dealt a heart.

Six stand-up comics, A, B, C, D, E, and F, are to perform on a single evening at a comedy club. The order of performance is determined by random selection. Find the probability that a. Comic E will perform first. b. Comic \(\mathrm{C}\) will perform fifth and comic B will perform last. c. The comedians will perform in the following order: D, E, C, A, B, F. d. Comic \(\mathrm{A}\) or comic \(\mathrm{B}\) will perform first.

Does the problem involve permutations or combinations? Explain your answer. (It is not necessary to solve the problem.) How many different four-letter passwords can be formed from the letters \(A, B, C, D, E, F\), and \(G\) if no repetition of letters is allowed?

A popular brand of pen is available in three colors (red, green, or blue) and four writing tips (bold, medium, fine, or micro). How many different choices of pens do you have with this brand?

The tables in Exercises 3-4 show claims and their probabilities for an insurance company. a. Calculate the expected value and describe what this means in practical terms. b. How much should the company charge as an average premium so that it breaks even on its claim costs? c. How much should the company charge to make a profit of \(\$ 50\) per policy? PROBABILITIES FOR MEDICAL INSURANCE CLAIMS $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Amount of Claim (to the } \\ \text { nearest } \$ 20,000) \end{array} & \text { Probability } \\ \hline \$ 0 & 0.70 \\ \hline \$ 20,000 & 0.20 \\ \hline \$ 40,000 & 0.06 \\ \hline \$ 60,000 & 0.02 \\ \hline \$ 80,000 & 0.01 \\ \hline \$ 100,000 & 0.01 \\ \hline \end{array} $$

You are dealt one card from a 52-card deck. Find the probability that you are not dealt a club.

Seven performers, A, B, C, D, E, F, and G, are to appear in a fund raiser. The order of performance is determined by random selection. Find the probability that a. D will perform first. b. E will perform sixth and \(B\) will perform last. c. They will perform in the following order: C, D, B, A, G, \(\mathrm{F}, \mathrm{E}\). d. \(F\) or \(G\) will perform first.

Does the problem involve permutations or combinations? Explain your answer. (It is not necessary to solve the problem.) Fifty people purchase raffle tickets. Three winning tickets are selected at random. If each prize is \(\$ 500\), in how many different ways can the prizes be awarded?

Use the Fundamental Counting Principle to solve Exercises 1-12. In how many different ways can a police department arrange eight suspects in a police lineup if each lineup contains all eight people?

In how many ways can a casting director choose a female lead and a male lead from five female actors and six male actors?

An architect is considering bidding for the design of a new museum. The cost of drawing plans and submitting a model is \(\$ 10,000\). The probability of being awarded the bid is \(0.1\), and anticipated profits are \(\$ 100,000\), resulting in a possible gain of this amount minus the \(\$ 10,000\) cost for plans and a model. What is the expected value in this situation? Describe what this value means.

You are dealt one card from a 52-card deck. Find the probability that you are not dealt a picture card.

A group consists of four men and five women. Three people are selected to attend a conference. a. In how many ways can three people be selected from this group of nine? b. In how many ways can three women be selected from the five women? c. Find the probability that the selected group will consist of all women.

In Exercises 5-20, use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{6} C_{5}\)

Use the Fundamental Counting Principle to solve Exercises 1-12. As in Exercise 1, six performers are to present their comedy acts on a weekend evening at a comedy club. One of the performers insists on being the last stand-up comic of the evening. If this performer's request is granted, how many different ways are there to schedule the appearances?

A student is planning a two-part trip. The first leg of the trip is from San Francisco to New York, and the second leg is from New York to Paris. From San Francisco to New York, travel options include airplane, train, or bus. From New York to Paris, the options are limited to airplane or ship. In how many ways can the two-part trip be made?

A construction company is planning to bid on a building contract. The bid costs the company \(\$ 1500\). The probability that the bid is accepted is \(\frac{1}{5}\). If the bid is accepted, the company will make \(\$ 40,000\) minus the cost of the bid. Find the expected value in this situation. Describe what this value means.

You are dealt one card from a 52-card deck. Find the probability that you are not dealt a red picture card.

A political discussion group consists of five Democrats and six Republicans. Four people are selected to attend a conference. a. In how many ways can four people be selected from this group of eleven? b. In how many ways can four Republicans be selected from the six Republicans? c. Find the probability that the selected group will consist of all Republicans.

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{8} C_{7}\)

Use the Fundamental Counting Principle to solve Exercises 1-12. As in Exercise 2, five singers are to perform at a night club. One of the singers insists on being the last performer of the evening. If this singer's request is granted, how many different ways are there to schedule the appearances?

For a temporary job between semesters, you are painting the parking spaces for a new shopping mall with a letter of the alphabet and a single digit from 1 to 9 . The first parking space is A1 and the last parking space is Z9. How many parking spaces can you paint with distinct labels?

It is estimated that there are 27 deaths for every 10 million people who use airplanes. A company that sells flight insurance provides \(\$ 100,000\) in case of death in a plane crash. A policy can be purchased for \(\$ 1\). Calculate the expected value and thereby determine how much the insurance company can make over the long run for each policy that it sells.

Mega Millions is a multi state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to 46. Use this information to solve Exercises 7-10. Express all probabilities as fractions. A player wins the jackpot by matching all five numbers drawn from the white balls ( 1 through 56 ) and matching the number on the gold Mega Ball \({ }^{\circledR}\) (1 through 46). What is the probability of winning the jackpot?

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{9} C_{5}\)

Use the Fundamental Counting Principle to solve Exercises 1-12. You need to arrange nine of your favorite books along a small shelf. How many different ways can you arrange the books, assuming that the order of the books makes a difference to you?

An ice cream store sells two drinks (sodas or milk shakes), in four sizes (small, medium, large, or jumbo), and five flavors (vanilla, strawberry, chocolate, coffee, or pistachio). In how many ways can a customer order a drink?

A 25 -year-old can purchase a one-year life insurance policy for \(\$ 10,000\) at a cost of \(\$ 100\). Past history indicates that the probability of a person dying at age 25 is \(0.002\). Determine the company's expected gain per policy.

Mega Millions is a multi state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to 46. Use this information to solve Exercises 7-10. Express all probabilities as fractions. A player wins a minimum award of \(\$ 10,000\) by correctly matching four numbers drawn from the white balls (1 through 56) and matching the number on the gold Mega Ball \({ }^{\circledR}\) (1 through 46). What is the probability of winning this consolation prize?

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{10} C_{6}\)

Use the Fundamental Counting Principle to solve Exercises 1-12. You need to arrange ten of your favorite photographs on the mantel above a fireplace. How many ways can you arrange the photographs, assuming that the order of the pictures makes a difference to you?

A pizza can be ordered with three choices of size (small, medium, or large), four choices of crust (thin, thick, crispy, or regular), and six choices of toppings (ground beef, sausage, pepperoni, bacon, mushrooms, or onions). How many onetopping pizzas can be ordered?

Full house: 3 cards of one number and 2 cards of a second number

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{11} C_{4}\)

A restaurant offers the following limited lunch menu.$$ \begin{array}{|l|l|l|l|} \hline \text { Main Course } & \text { Vegetables } & \text { Beverages } & \text { Desserts } \\ \hline \text { Ham } & \text { Potatoes } & \text { Coffee } & \text { Cake } \\\ \hline \text { Chicken } & \text { Peas } & \text { Tea } & \text { Pie } \\ \hline \text { Fish } & \text { Green beans } & \text { Milk } & \text { Ice cream } \\ \hline \text { Beef } & & \text { Soda } & \\ \hline \end{array} $$ If one item is selected from each of the four groups, in how many ways can a meal be ordered? Describe two such orders.

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{12} C_{5}\)

A store specializing in mountain bikes is to open in one of two malls. If the first mall is selected, the store anticipates a yearly profit of \(\$ 300,000\) if successful and a yearly loss of \(\$ 100,000\) otherwise. The probability of success is \(\frac{1}{2}\). If the second mall is selected, it is estimated that the yearly profit will be \(\$ 200,000\) if successful; otherwise, the annual loss will be \(\$ 60,000\). The probability of success at the second mall is \(\frac{3}{4}\). Which mall should be chosen in order to maximize the expected profit?