Chance and risk play a role in everyone s life. No

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1 CAPER Counting 6 and Probability Lesson 6.1 A Counting Activity Chance and risk play a role in everyone s life. No doubt you have often heard questions like What are the chances? Some risks are avoidable, but others are not. For example, everyone lives with the risk of catching a cold, but people do not have to play gambling games such as lotteries. he popularity of lotteries attests to the fascination that people have with risk: In 2011, Americans spent $251 per person on lotteries. Often questions of probability are associated with questions about counting. In order to determine the probability of winning a lottery jackpot, a mathematician must calculate the number of ways of selecting numbers from those that are available. In how many ways can a lottery participant choose several numbers from those on a lottery ticket? What is the probability of winning a lottery jackpot? What is the probability that a medical test s results are correct? ow has an understanding of probability helped improve the reliability of U.S. space shuttle launches? In this chapter, you will examine a variety of questions about counting and probability that are fundamental to modeling random events. Lesson 6.2 Counting echniques, Part 1 Lesson 6.3 Counting echniques, Part 2 Lesson 6.4 Probability, Part 1 Lesson 6.5 Probability, Part 2 Chapter Extension Monte Carlo Models Chapter 6 Review

2 Lesson 6.1 A Counting Activity Probability calculations are important in many modeling situations. A meteorologist, for example, must calculate the probability of rain, a lottery commission must calculate the probability a player will win, and a medical researcher must calculate the probability that the results of tests are correct. Many probability calculations require knowing the number of ways in which an event can happen, such as the number of ways a lottery player can fill out a lottery card. Often the numbers involved are quite large, and careful methods must be used to be sure counting is done properly. owever, the best way to begin your work is by considering some situations involving relatively small numbers. he Wizard of Id by Brant Parker and Johnny art. Reprinted by permission of John L. art FLP, and Creators Syndicate, Inc.

3 Lesson 6.1 A Counting Activity 299 Explore his he Central igh School student council is discussing three fundraising proposals. Pierre suggests that the council operate a game at the annual school fair. is idea is to write each of the letters of the school s team name, Lions, on a Ping-Pong ball and have participants draw two of the balls from an opaque container. If the letters spell (in the order drawn) a legal word, the participant wins a prize. Some council members are critical of the idea because they think the game is too easy to win. ilary proposes printing cards with the numbers 1 through 9 displayed in a square matrix and having participants mark two of the numbers (see Figure 6.1). A winning pair is generated at random, and a prize is given to any participant who matches both winning numbers. er scheme leaves council members uncertain about how many winners can be expected in a school of 1,000 students Figure 6.1. A card for ilary s game. Chuck also wants to operate a game at the school fair. is game involves a board with the numbers 1 through 6 displayed (see Figure 6.2). A participant places $1 on any of the numbers, rolls two dice, and wins a dollar for each time the chosen number appears. Several council members feel that the organization would lose money on this game Figure 6.2. he board for Chuck s game. Following are three sets of questions related to the games suggested by Pierre, ilary, and Chuck. As time permits, discuss one or more of the three sets with a few other people.

4 300 Chapter 6 Counting and Probability ere is one way to divide the sets of questions among small groups in your class. At the direction of your instructor, divide your class into groups of three people. Write the numbers 1, 2, or 3 on each of several slips of paper. ave each group draw one of the slips from a bag or box. Each group should consider the set of questions whose number corresponds to the number drawn. 5,631 People Win N.Y. Lotto with 911 on 9/11 wnbc.com September 14, 2002 On the first anniversary of the terrorist attacks on New York City, a date often referred to as simply 9/11, the evening numbers drawn in the New York Lottery were 911. Lottery officials said hursday that 5,631 people selected the winning sequence. hey will each win $500. Others won with box bets. he liability limit for the midday and evening draws is $10,000 in sales to ensure a maximum payout. he maximum for a straight bet, or matching the exact sequence of winning numbers, is $500 for a $1 ticket. Bettors can also win $250 on a 50-cent ticket. On any given day, 7 10 sets of numbers are closed out. By 5:09 p.m. uesday, the combination for both midday and evening draws Wednesday had reached its limit. According to past winning numbers listed on the Lottery Website, Wednesday was the first time in more than a year that the 911 combination came up. A similar coincidence occurred Nov. 12 when the numbers came up in the New Jersey Lottery the day American Airlines Flight 587 crashed on the New York coast. After all groups have finished their discussions, a spokesperson for each group should present the results of the group s discussion to the class. he groups that discussed set 1 should report first, and so forth. 1. Analyze Pierre s proposal. ow many different two-letter words are there? ow many of them are real words? If each of the school s 1,000 students enters exactly once and pays a $1 entry fee, how many winners might there be? ow much should each winner receive if the council hopes to raise $500? 2. Analyze ilary s proposal. In how many ways can a student fill in the entry form? If each of the school s 1,000 students enters exactly once and pays a $1 entry fee, how many winners might there be? ow much should each winner receive if the council hopes to raise $500? 3. Analyze Chuck s proposal. In how many ways can the two dice fall? ow often would the council pay the participant $1? $2? ow often would the council make $1? Do you think that the council can raise $500 if that is the goal? (If you want to try the game, check with your teacher to see if dice are available in your classroom.)

5 Lesson 6.1 A Counting Activity 301 Exercises 1. One of the goals of this chapter is to develop a few techniques that can be used to determine the number of ways in which an event can happen. he most basic such technique is making a list of all possible ways. his is a reasonable method as long as the number of items in the list is not too large. Make a list of all possible words that can be made by using two letters of the word Lions. 2. Suppose the Ping-Pong balls in Pierre s game are drawn one at a time, and the first is kept out of the container while the second is drawn. ow many different letters could appear on the first Ping- Pong ball? he second? What is the connection between these numbers and the number of words in the list you made in Exercise 1? If the school s team name were igers, how many words of two letters would there be? 3. Make a list of all possible ways of choosing a pair of numbers from the nine available on one of ilary s cards. ow many are there? 4. If you are filling in one of ilary s cards, in how many ways can you select your first number? After you ve picked your first number, in how many ways can you pick your second number? ow are these two numbers related to the number of pairs you listed in Exercise 3? 5. Since Chuck s game involves two dice, it is important to be able to distinguish them. herefore, imagine the dice are different colors, say red and green. One way the dice can fall is the red die a 3 and the green die a 4. his can be written in shorthand as the pair (3, 4). his outcome is different from the red die a 4 and the green die a 3, which can be written as (4, 3). Make a list of all possible ways the red die and the green die can fall together. ow many pairs are in your list? 6. he red die can land in six different ways, as can the green. ow are these two sixes related to the number of things in the list you made in Exercise 5?

6 302 Chapter 6 Counting and Probability A second counting technique is known in mathematics as the fundamental multiplication principle. It says that if events A and B can occur in a and b ways separately, then there are a b ways that the events can occur together. o use the principle, make a blank for each event and write the number of ways each event can occur in a blank. hen multiply these numbers. For example, to determine the number of ways that a die and a coin can fall together, make two blanks:. hen write the number of possibilities for the die and the coin in the blanks: 6 2, and multiply to get 12. If a full list of the 12 is needed, a systematic way to ensure that all items are listed is to make a tree diagram like that shown here. Die Coin Outcome 1, 1 1, 2 2, 2, 3 3, 3, 4 4, 4, 5 5, 5, 6 6, 6, 7. Explain how the multiplication principle can be applied to find the number of different words of two letters that can be made from the letters of Lions.

7 Lesson 6.1 A Counting Activity A utility company in North Dakota once sponsored a contest to promote energy conservation. he contest was to find all legal words that can be made from the letters of insulate without using any letter more than once. a. Use the multiplication principle to determine the number of words of two letters that are possible. b. he multiplication principle can be extended to three or more events. Show how to apply the principle to determine the number of words that can be made from three letters of insulate. 9. It is possible to modify the multiplication principle to find the number of ways of selecting two numbers on one of ilary s cards. Explain how this can be done. 10. Why is it necessary to modify the multiplication principle in Exercise 9? 11. Lotteries often require the participant to select several numbers from a collection of numbers printed on a card. If a state lottery has the numbers 1 through 25 printed in a square matrix, in how many ways can a participant select two of them? Explain. 12. Explain how the multiplication principle can be used to determine the number of ways in which two dice can fall. 13. Make a tree diagram to show all the outcomes when a red die and a green die are tossed together. 14. Make a tree diagram to show all the possibilities when filling out one of ilary s cards. 15. You are playing Chuck s game and decide to bet on the number 5. a. Use the tree diagram you made in Exercise 13 to count the number of ways in which you can win or lose. In how many ways can you win $1? $2? In how many ways can you lose $1? b. If you played many times, do you think you would win or lose money in the long run? Explain. 16. In a common carnival dice game, three dice are rolled. Use the multiplication principle to determine the number of ways in which three dice can fall.

8 304 Chapter 6 Counting and Probability 17. Read the news article on page 300. Use the counting techniques you learned in this lesson to find the number of ways that a player can place a bet in the New York lottery game described in the article. 18. Counting techniques are useful in many modeling situations other than the analysis of games. An example is genetics. As you may know from your study of biology, a female inherits an X chromosome from her mother and another X chromosome from her father. A male inherits an X chromosome from his mother and a Y chromosome from his father. Use the counting techniques you learned in this lesson to explain the different ways in which chromosomes can be passed from parents to offspring. Computer/Calculator Explorations 19. Many calculators have built-in random-number generators that can be modified to simulate random situations. Adapt the randomnumber generator of a calculator to simulate the games proposed by Pierre, ilary, and Chuck. Present your work to the members of your class. Projects 20. Research and report on the impact of lotteries in the United States. What are the benefits and problems associated with lotteries?