S E C T I O N

7.12

Carnivals often have beanbag toss games, in which the contestant throws a beanbag at a board with holes cut in it and wins a prize if the beanbag goes through the hole. Variations of beanbag toss games are played at children's birthday parties and at tailgate parties before football games.

GAME 1 Beanbag Toss

*

*

Some people are skilled at beanbag tosses, and others are not. Ramona is very skilled at tossing the beanbag into the hole but her friend Ralph is not. At their high school carnival, they decide to play the beanbag toss game, the board for which is shown in Figure 7.46. Ramona throws the first beanbag and wins a prize immediately. Ralph, on the other hand, decides to randomly throw the beanbag toward the board. What is the probability that the beanbag tossed by Ralph will go through a hole in the board?

1. What is a favorable outcome for Ralph in this situation?

2. What are the possible outcomes in this situation?

3. Thinking geometrically, what do we call the measure of the interior region in a plane that an object covers?

FIGURE 7.46 4. How could you determine the size of the region that could result in a favorable outcome?

5. How could you determine the size of the region that could result in any possible outcome?

6. The game board is 5 feet wide and 6 feet long. Since there are 12 inches in a foot, what is the area of the board in square inches?

Section 7.12 Chapter 7 What Are the Chances? Probability 483

7. Each circular hole has a diameter of 5 inches. What is the radius of each hole?

8. What is the area of each circular hole?

9. What is the total area of the circular holes in the board?

10. What would be the probability of the beanbag landing in any one of the holes? (Assume that the bag is thrown at random and lands on the board.)

GAME 2: Spin the Spinner

In Section 7.2 you used a spinner to make predictions from experimental probability. The next carnival game that Ramona and Ralph encountered is a spinner game. For one ticket, they can spin the spinner and win the prize written in the section where the arrow lands. The spinner is shown in Figure 7.47.

Ralph may not be good at the beanbag toss but he does know geometry. He estimates the measures of the central angles for the various sectors of the spinner to be: 120° for the sticker set, 60° for the necklace, 45° for the watch, 90° for the rubber snake, and 45° for the ring.

FIGURE 7.47. Spinner game.

1. Suppose that Ralph wants to win the rubber snake. If each degree measure of the central angle is an outcome, how many favorable outcomes could there be to win the rubber snake? How do you know?

4 8 4 Chapter 7 What Are the Chances? Probability Section 7.12

2. How many outcomes would there be in the entire spinner? How do you know?

3. What would be the probability of winning the rubber snake?

4. Suppose that Ramona wants to win a piece of jewelry. How many possible favorable outcomes are there?

5. What would be the probability of winning a piece of jewelry?

6. What would be the probability of winning anything but the sticker set? Explain how you got your answer.

G A M E 3:Tangram Puzzle

Tangrams are ancient Chinese geometric puzzles with seven tiles that can fit together in a variety of ways. One of the carnival games has contestants toss a small beanbag into a box that hides a set of tangrams (Figure 7.48). A prize is awarded depending on where the beanbag lands.

1. The entire tangram square is four units wide and four units long. In terms of square units, what is the area of the large blue triangle?

2. In terms of square units, what is the area of the small purple triangle?

3. What is the probability of the beanbag landing on each of the following pieces?

a. Pink parallelogram

b. Orange square

c. Green triangle

4. Suppose that the box is knocked over and the tangrams fall out of their frame. If they are reassembled in a different arrangement that still makes the large square (assume that the pieces do not overlap and there are no gaps), do the probabilities of the beanbag landing on any given piece change? Explain your answer.

FIGURE 7.48

Section 7.12 Chapter 7 What Are the Chances? Probability 485

SECTION

Use a paper clip and pencil to create a spinner using the circle and sectors shown in Figure 7.50.

1. Spin the spinner twenty times (be sure to spin the same way each time) and record your results in Figure 7.49.

Area 1 Area 2 Area 3 | Area 4

Tally marks

Total:

Relative frequency (total + 20)

FIGURE 7.49

FIGURE 7.50

486 Chapter 7 What Are the Chances? Probability Assignment 7.12

2. Use the data from your chart to create a bar graph of your data on a grid like Figure 7.51.

20

16

16

'4

12

10

8

6

4

2

FIGURE 7.51

3. Use a protractor and measure the central angle for each sector of the circle (Figure 7.52).

Area 1 Area 2 Area 3 Area 4

Measure of Central Angle

Fercent of Circle (measure of central angle - 560°)

FIGURE 7.52

4. Compare the relative frequencies of the spins from your chart in Question 1 and the percentage of circle in your chart in Question 3. How close are the numbers? Should they be close? Explain.

Extend What You Know!

5. Create a spinner with the following sectors and percentages of the whole circle:

Red 15% = 54° Blue 20% =72° Yellow 10% = 36° Green 55% = 198°

Spin the spinner twenty times. Record the same data you recorded in the previous questions. Are your experimental results close to the assigned values? List some reasons for errors. Turn in both the actual spinner and the lab write-up.

Assignment 7.12 Chapter 7 What Are the Chances? Probability 487