A C E. Answers Investigation 3. Applications. 12, or or 1 4 c. Choose Spinner B, because the probability for hot dogs on Spinner A is

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1 Answers Investigation Applications. a. Answers will vary, but should be about for red, for blue, and for yellow. b. Possible answer: I divided the large red section in half, and then I could see that the spinner has six equal sections. The red occupies, or ; the blue occupies, or ; and the yellow occupies. c. Possible answer: These answers are not likely to be exactly the same, but if we do many trials they should be close. d. Possible answer: My answers should be closer to the actual fractions, because I would be using more data to determine them. e. No; the different colors make up different degrees of the total angle of the spinner. For example, there is a greater chance of landing on a space with red than on a space with blue, because the angles associated with red (80 ) are greater than the angles associated with blue (0 ) or yellow (0 ). f. Possible answers: I could score a point on both blue and yellow to make the game fair. This would give half the area to each of us. Or, I could score three points for landing on yellow, and neither of us could score when the spinner lands on blue. g. Possible answer: In spins, Player A would expect to score about times, Player B would expect to score about 8 times, and Player C would expect to score about times. To make the points that each player would accumulate the same, you could let Player A score points, Player B score points, and Player C score points; then each would expect points for spins.. a. You could choose either spinner, because Spinner A and Spinner B both have a probability of, or, for landing on pizza. b. Choose Spinner A, because the probability for lasagna on Spinner A is, and on Spinner B it is, or. or c. Choose Spinner B, because the probability for hot dogs on Spinner A is or, and on Spinner B it is, or.. a. Not equally likely; Region is more likely. Since Region has a bigger central angle associated with it than Region does, it has a greater theoretical probability. b. Equally likely; both sections have sides that make a 80 angle with the pointer s fixed point, or center of rotation.. The third data set ( ones, twos, and 5 threes) is from Spinner B. Students might argue that 0 trials are not enough to be certain which spinner generated which data set, and this is certainly true. Based on the spinners, it appears that the first data set is from Spinner C and the second data set is from Spinner A. 5. a. Spinner A; each player has the same amount of area, or central angle, in which to score. b. Possible answers: On Spinner B, the section makes up of the area, the sections make up, or, of the area, and the section makes up of the area. In four spins, Player can expect to score once, Player can expect to score twice, and Player can expect to score once. To make the game fair, you can give two points to Player if the pointer lands on, one point to Player if the pointer lands on, and two points to Player if the pointer lands on. In eight spins, Player can expect to score twice for a total of points = points, Player can expect to score four times for a total of point = points, and Player can expect to score twice for a total of points = points. What Do You Expect? Investigation

2 Answers Investigation. B 7. F On Spinner C, the sections make up, or, of the area, the sections make up, or, of the area, and the section makes up of the area. In spins, Player can expect to score twice, Player can expect to score three times, and Player can expect to score once. To make the game fair, you can give three points to Player if the pointer lands on, two points to Player if the pointer lands on, and six points to Player if the pointer lands on. In spins, Player can expect to score four times for a total of points = points, Player can expect to score six times for a total of points = points, and Player can expect to win twice for a total of points = points. Connections 8. a. Answers will vary. For a fair game, the explanation should give the chances for each outcome and show that the expected number of points for each player is the same over a reasonable number of spins. 9. A 0. G. A b. Answers will vary. For a not-so-fair game, the same sort of example should be given for points expected for each player over some fixed number of spins.. J Note: Carlos has not chosen a strategy that is likely to lead to frequent success, but since Ella cannot possibly win, then Carlos has a 00% chance of being the eventual winner.., or ; six out of the possible outcomes are doubles.. Fifteen equally likely combinations are possible: AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, and EF. Of these, one is a matching pair, so the probability of winning is Fraction Decimal Percent % 0.5.5% % design is below. Lasagna should occupy of the circle, Hamburger should occupy of the circle, and Tuna should occupy of the circle , or about %. a. Students will have various ways of choosing this spinner. One possible What Do You Expect? Investigation

3 Answers Investigation b. 5%. Part (c) does not have the same probability as the others. The probability of getting one heads and one tails when you toss two coins is. Each of the others gives a probability of for spinach.. since 8 is equal to, 7 7 8, and since = 0., which is greater than a..% b..% c..% d. Answers will vary. Students who base their answers solely on the percentages will choose Player A (this is like looking at experimental probabilities of other people s data). Students may have good reasons for other answers (for example, Player B has more experience). Do not ignore real-life concerns that may appear nonmathematical. Students need to use mathematics sensibly along with their knowledge about other aspects of their lives to make good decisions. 7. a. No; the graph starts at 95%, so the bars are not proportional. b. Answers will vary. There are many relevant things you don t know. For example, you don t know how many trucks of each company were considered, or if each company has been making trucks for ten years. This last issue is important, because if Company A has manufactured most of its vehicles during the most recent five years, you would expect a higher percentage of Company A s trucks to still be on the road Students may argue that doubling 50% would give 00%, which means that it is a certainty that at least one even number would appear. But we know that we can get (, ), (, ), and so on. So 00% cannot be correct. Or, they may make a tree or list to show all the possibilities and count the outcomes that have at least one even number. 7 out of outcomes have at least one even number.. a. This pyramid would have four faces (a tetrahedron). b. If all faces are congruent, then the probability of the shaded face landing on the bottom is. c. You can make a pyramid with this, but the square base of the shape is larger than the other faces so the shape is likely to end up on that face more often. This would mean that all outcomes are not equally likely. This would make for an unfair game.. 5 9, or. 5, or ; the possible prime sums are,, 5, 7, and. There is one way to get a sum of, two ways to get a sum of, four ways to get a sum of 5, six ways to get a sum of 7, and two ways to get a sum of. So, there are = 5 ways to get a prime number.. The probability of getting a sum that is a factor of is greater. There are five ways to get, two ways to get, and one way to get, so the probability of getting a sum that is a factor of is 8, or 9. There are five ways to get and one way to get, so the probability of getting a sum that is a multiple of is, or What Do You Expect? Investigation

4 Answers Investigation 7. a. This table lists the possible products. Cube Cube This table shows how many ways each product can occur. Product Number of Ways b. There are = 9 ways to get an odd product. The probability Nina wins is 9. There are = 7 ways to get an even product, so P(Humberto wins) = 7. Students might recognize there is an alternative way to find these probabilities. The only way to get an odd product is with two odd factors. The probability of an odd product then is. To find the probability of an even product, students could use the complement, so the probability of an even product is It is not necessary to mention this if students do not come up with it on their own. c. This is not a fair game of chance, because Humberto has a better chance of winning than Nina does. To make the game fair, you could give Nina points each time she has a chance to score. This will mean that in a game of rolls, Nina would expect to score times, or times for a total of 9 points. Humberto would expect to score times, or 9 times for a total of 9 points. d. e., or ;the factors of are,, and. There is one way to get, two ways to get, and three ways to get, so there are + + = ways to get a factor of. Note: Some students may focus on the number of even products and the number of odd products rather than on the number of ways to get an odd or an even product; that is, there are only 8 different products, odd and even. If you find your students making this mistake, discuss the question with them, emphasizing that some products can be generated in more than one way. What Do You Expect? Investigation

5 Answers Investigation 8. a. Answers will vary. Every card may win something, but the value of what is won may be very small, for example, a stick of gum, so that winning might not be winning something of significant value to the winner. Regardless of whether students think the ad is legal, they need to give a reasonable explanation for their response. b. Answers will vary. An ad mentioning big prizes might be more exciting. c. Answers will vary. Students could bring ads from very diverse companies. Examine the ad to determine if it suggests that a contestant should expect to win. It is important that students are able to explain why this ad is deceptive and to suggest how the company could improve its ad. 9. Possible answer: No; most dentists probably do not recommend chewing gum at all, but feel that if their patients are going to chew gum anyway, it should be sugarless. 0. a. 50%; the students could flip a coin with heads = yellow and tails = green. The students should conduct at least 5 trials. Students can record what percent landed on yellow, but they cannot evaluate the chance that 70% landed on yellow from one run. b. This simulation shows more insects on the yellow part, but the percentages, 5% and 8%, are very close. This can be an experimental result that is not exactly like the theoretical result. Extensions... Answers may vary: One possibility is to put 9 regular table tennis balls in a container and add one ball with an invisible mark that only shows up under a special light. Out of the 0 balls, only that one wins.. Answers may vary: Every customer that goes into any of the stores in the chain gets a receipt. The receipt number is recorded in a computer linked to all the stores. At the end of the contest, the computer randomly chooses a receipt and the stores have a winner. 5. Answer may vary: The auto dealer puts a tub with 500 tiny toy cars in front of his store. Each person gets to choose a car. One out of every ten cars in the tub has a small dot on a tire. The people who draw a car with a dot win $500. What Do You Expect? 5 Investigation