Chapter 11

In Exercises 11-14, a single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.

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 box contains 25 transistors, 6 of which are defective. If 6 are selected at random, find the probability that a. all are defective. b. none are defective.

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{8} C_{1}\)

Shoppers in a large shopping mall are categorized as male or female, over 30 or 30 and under, and cash or credit card shoppers. In how many ways can the shoppers be categorized?

An oil company is considering two sites on which to drill, described as follows: Site A: Profit if oil is found: \(\$ 80\) million Loss if no oil is found: \(\$ 10\) million Probability of finding oil: \(0.2\) Site B: Profit if oil is found: \(\$ 120\) million Loss if no oil is found: \(\$ 18\) million Probability of finding oil: \(0.1\) Which site has the larger expected profit? By how much?

In Exercises 11-14, a single die is rolled twice. Find the probability of rolling a 5 the first time and a 1 the second time.

A committee of five people is to be formed from six lawyers and seven teachers. Find the probability that a. all are lawyers. b. none are lawyers.

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{7} C_{1}\)

There are three highways from city A to city \(B\), two highways from city \(B\) to city \(C\), and four highways from city \(C\) to city D. How many different highway routes are there from city A to city D?

In a product liability case, a company can settle out of court for a loss of \(\$ 350,000\), or go to trial, losing \(\$ 700,000\) if found guilty and nothing if found not guilty. Lawyers for the company estimate the probability of a not-guilty verdict to be \(0.8\). a. Find the expected value of the amount the company can lose by taking the case to court. b. Should the company settle out of court?

In Exercises 11-14, a single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 2 the second time.

A city council consists of six Democrats and four Republicans. If a committee of three people is selected, find the probability of selecting one Democrat and two Republicans.

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

In Exercises 13-32, evaluate each factorial expression. \(\frac{9 !}{6 !}\)

A person can order a new car with a choice of six possible colors, with or without air conditioning, with or without automatic transmission, with or without power windows, and with or without a CD player. In how many different ways can a new car be ordered with regard to these options?

A service that repairs air conditioners sells maintenance agreements for \(\$ 80\) a year. The average cost for repairing an air conditioner is \(\$ 350\) and 1 in every 100 people who purchase maintenance agreements have air conditioners that require repair. Find the service's expected profit per maintenance agreement.

In Exercises 11-14, a single die is rolled twice. Find the probability of rolling an odd number the first time and a number less than 3 the second time.

A parent-teacher committee consisting of four people is to be selected from fifteen parents and five teachers. Find the probability of selecting two parents and two teachers.

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

Evaluate each factorial expression. \(\frac{12 !}{10 !}\)

A car model comes in nine colors, with or without air conditioning, with or without a sun roof, with or without automatic transmission, and with or without antilock brakes. In how many ways can the car be ordered with regard to these options?

Exercises 15-19 involve computing expected values in games of chance. A game is played using one die. If the die is rolled and shows 1 , the player wins \(\$ 5\). If the die shows any number other than 1 , the player wins nothing. If there is a charge of \(\$ 1\) to play the game, what is the game's expected value? What does this value mean?

In Exercises 15-20, you draw one card from a 52-card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of drawing a picture card the first time and a heart the second time.

Exercises 15-20 involve a deck of 52 cards. If necessary, refer to the picture of a deck of cards, as shown in Figure 11.5 on page 718 . A poker hand consists of five cards. a. Find the total number of possible five-card poker hands. b. A diamond flush is a five-card hand consisting of all diamonds. Find the number of possible diamond flushes. c. Find the probability of being dealt a diamond flush.

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{30} C_{3}\)

Evaluate each factorial expression. \(\frac{29 !}{25 !}\)

You are taking a multiple-choice test that has five questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions?

Involve computing expected values in games of chance. A game is played using one die. If the die is rolled and shows 1 , the player wins \(\$ 1\); if 2 , the player wins \(\$ 2\); if 3 , the player wins \(\$ 3\). If the die shows 4,5 , or 6 , the player wins nothing. If there is a charge of \(\$ 1.25\) to play the game, what is the game's expected value? What does this value mean?

You draw one card from a 52-card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of drawing a jack the first time and a club the second time.

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{25} C_{4}\)

Evaluate each factorial expression. \(\frac{31 !}{28 !}\)

You are taking a multiple-choice test that has eight questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions?

Involve computing expected values in games of chance. Another option in a roulette game (see Example 6 on page 760 ) is to bet \(\$ 1\) on red. (There are 18 red compartments, 18 black compartments, and 2 compartments that are neither red nor black.) If the ball lands on red, you get to keep the \(\$ 1\) that you paid to play the game and you are awarded \(\$ 1\). If the ball lands elsewhere, you are awarded nothing and the \(\$ 1\) that you bet is collected. Find the expected value for playing roulette if you bet \(\$ 1\) on red. Describe what this number means.

You draw one card from a 52-card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of drawing a king each time.

In Exercises 17-22, you randomly select one card from a 52-card deck. Find the probability of selecting a 2 or a 3 .

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{5} C_{0}\)

Evaluate each factorial expression. \(\frac{19 !}{11 !}\)

In the original plan for area codes in 1945, the first digit could be any number from 2 through 9 , the second digit was either 0 or 1 , and the third digit could be any number except 0 . With this plan, how many different area codes are possible?

Involve computing expected values in games of chance. The spinner on a wheel of fortune can land with an equal chance on any one of ten regions. Three regions are red, four are blue, two are yellow, and one is green. A player wins \(\$ 4\) if the spinner stops on red and \(\$ 2\) if it stops on green. The player loses \(\$ 2\) if it stops on blue and \(\$ 3\) if it stops on yellow. What is the expected value? What does this mean if the

You draw one card from a 52-card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of drawing a 3 each time.

You randomly select one card from a 52-card deck. Find the probability of selecting a 7 or an \(8 .\)

Involve a deck of 52 cards. If necessary, refer to the picture of a deck of cards, as shown in Figure 11.5 on page 718 . If you are dealt 4 cards from a shuffled deck of 52 cards, find the probability that all 4 are hearts.

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{6} C_{0}\)

Evaluate each factorial expression. \(\frac{17 !}{9 !}\)

The local seven-digit telephone numbers in Inverness, California, have 669 as the first three digits. How many different telephone numbers are possible in Inverness?

Involve computing expected values in games of chance. For many years, organized crime ran a numbers game that is now run legally by many state governments. The player selects a three-digit number from 000 to 999 . There are 1000 such numbers. A bet of \(\$ 1\) is placed on a number, say number 115 . If the number is selected, the player wins \(\$ 500\). If any other number is selected, the player wins nothing. Find the expected value for this game and describe what this means

You draw one card from a 52-card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of drawing a red card each time.

You randomly select one card from a 52-card deck. Find the probability of selecting a red 2 or a black 3 .

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \(\frac{{ }_{7} C_{3}}{{ }_{5} C_{4}}\)

Evaluate each factorial expression. \(\frac{600 !}{599 !}\)