1 Answers Investigation ACE Assignment Choices Problem. Core, 6 Other Connections, Extensions Problem. Core 6 Other Connections 7 ; unassigned choices from previous problems Problem. Core 7 9 Other Connections 5 ; Extensions 5, 6; unassigned choices from previous problems Problem. Core 0 Other Connections, ; Extensions 7; unassigned choices from previous problems Adapted For suggestions about adapting Exercise 7 and other ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units, 6, 5 8: Bits and Pieces I;, 5, 0, : Bits and Pieces II; 9: Data About Us Applications. a. P(green) = P() =, or P() = b. + + = c. ; three of the four blocks are not. d. + =. a. P(green) = 5 6 P() = 5 P(orange) = 5 5 P() =, or 6 5 b = c. P(green) = 8% P() = % P(orange) = 8% P() = 0% d. 8% + % + 8% + 0% = 00% e. or 00%; Possible explanation: If the possible outcomes of an action do not overlap and account for everything that might happen, then the sum of the probabilities of the outcomes will be, or 00%.. a. P(white) =, or 6 P() = 6 P() =, or 6 b. ; the probability of choosing a white block is, so the probability of not choosing a white block is =. c. The probabilities are the same. There will be white blocks, blocks, and 6 blocks, so P(white) = = ; P() =, or ; 6 6 P() =, or. d. P(white) =, or P() =, or 5 P() = e. Answers may vary. Possible answer: Add blocks. Then there are 5 blocks, white blocks, and blocks, for a 5 total of 0 blocks. And P() =, or a. = b. 0 c. = 0 5. a. Since the probability of choosing a white marble is, there must be at least 0 0 marbles in the bag. Investigation Experimental and Theoretical Probability 5 ACE ANSWERS
2 b. Yes. Possible answer: Let s assume the bag contains 60 marbles. Since the probability of choosing is, then of the proposed 60 marbles, or marbles, would be. Since of the 60 marbles must be white, 0 8 would be white. This gives us a total of 0 marbles, so the other 0 marbles would be. Since we get whole numbers of marbles when we do this division, the bag could contain 60 marbles. c. Possible answer: Since of the marbles in 0 the bag are white, and 6 marbles are white, we need to answer this question: of what 0 number equals 6? The answer is 0. This works because P() =, and of 0 = marbles. Since marbles are and 6 are white, there are 0 0 = 0 marbles in the bag. d. Possible answer: You could tell by adding the other two probabilities (of and white) and subtracting the result from : 5 + = + =, and =, or. So the probability of choosing a marble is. 6. a. True. The outcome must be impossible (such as rolling a 7 on a number cube). Figure Sandwich Chicken Hamburger Turkey Vegetable b. True. The outcome must be absolutely certain (such as rolling a number less than 7 on a number cube). c. False. All probabilities are between 0 (impossible) and (absolutely certain), inclusive. 7. a. For paper color followed by marker color, there are outcomes. Fruit Paper Color pink green Marker Color Outcome Outcome pink- pink- pink- green- green- green- Chicken-- Chicken-- Chicken-- Chicken-- Hamburger-- Hamburger-- Hamburger-- Hamburger-- Turkey-- Turkey-- Turkey-- Turkey-- 5 How Likely Is It?
3 b. P(pink-) =. c. P( paper) =. P(not paper) = or P( paper), which is =. d. P( marker) =, or. 8. a. There are different possible lunches. (Figure, previous page) b. The probability of Sol getting his favorite lunch is. Since the cook is not paying any attention to how she puts the lunches together, and there are an equal number of each kind of sandwich, vegetable, and fruit, each of the combinations is equally likely. c. The probability of Sol getting at least one 0 5 of his favorite things is, or. Only of 6 the combinations of items [(hamburger, spinach, apple) and (turkey, spinach, apple)] don t contain at least one of his favorite things. 9. a. The possible outcomes of a spin followed by a roll of the number cube are: (, ), (, ), (, ), (, ), (, 5), (, 6), (, ), (, ), (, ), (, ), (, 5), and (, 6). b. Since (, ) is one of equally likely possibilities, the probability is. c. Since the factors of are and, the only possibilities are (, ), (, ), (, ), and (, ). Thus, there are ways out of equally likely outcomes, so the probability is, or. The multiples of in the data are,, and 6, so the only possibilities are (, ), (, ), and (, 6). Thus, there are ways out of equally-likely outcomes, so the probability is =. 0. The outcomes are the same for the two situations. We can see this by identifying the three coins in the first case with the three tosses in the second. Coin can be heads or tails, just as the first toss can be, and so on. Tossing a coin three times or tossing three coins at once does have the same number of equally likely outcomes which include: HHH, TTT, THT, HTH, TTH, HHT, THH, and HTT. Note: Some students may answer no for this question, which is fine as long as their reasoning is correct. They may say that the outcomes for tossing three coins at once are: three heads, three tails, two heads and one tail, or two tails and one head. These descriptions do describe the outcomes, though notice that these outcomes are not equally likely. This must be the case when determining theoretical probability.. This is not a fair game. There are two winning outcomes for Eva (THT and HTH), but four winning outcomes for Pietro (TTH, HHT, THH, and HTT). Two outcomes (HHH and TTT) have no winners. The game can be made fair by changing the point scheme. Eva can be awarded twice as many points as Pietro for each winning combination.. a. Odd-Odd, Odd-Even, Even-Odd, Even-Even b. Possible answer: If the sum of the two number cubes is even, Player scores a point. If the sum of the two number cubes is odd, Player scores a point. c. Possible answer: If the product of the two number cubes is even, Player scores a point. If the product of the two number cubes is odd, Player scores a point. d. As with tossing a coin, considering even and odd on a number cube gives two equally likely outcomes. The game suggested in part (b) is like the game in Problem., One More Try. In this game, a match is two evens or two odds, whose sum is even. A no-match is one of each, with an odd sum. The games that students might imagine for part (b) might not be as easily connected to the two coins game. Connections 5. a. = = b. 7 = = 6.5 c. = = ACE ANSWERS Investigation Experimental and Theoretical Probability 55
4 . Parts (a) and (b) are both equal to. 5. Possible answer: For (a), if you are choosing one marble out of a bag that has, and white marbles, then the sum represents the sum of all the probabilities for each color. 6. a. Bly s probability is closer to. If students convert the fractions into decimal form they will get that Bly s probability is 0.75 and Kara s probability is about 0.7. b. Possible answers: Tossing a number cube and finding the probability that you will roll a number greater than. Choosing a block from a bag containing one, one, and one green block. 7. Answers will vary. Students answers should be fractions between and including 0 and (or percents between and including 0 and 00), and their reasoning should justify their answer. 5. A 6. H 7. B 8. G 9. a. From counting, we know there are 8 students in the class. Since choosing each student is equally likely, and of the 8 have first names that begin with J, the probability is, or. 8 7 b. There are 7 names that begin with a letter from G through S, so the probability of 7 choosing a student in this range is. 8 c., because there is only one person in the 8 class whose first name begins with K. d. The class now has 0 students, and since there are still only students whose names begin with J, the new probability is = a. You need to figure to figure the 7 probability of white. You need to find + + and subtract from to figure 7 the probability of. b. The total number of marbles has to be a multiple of, 7, and since these are the denominators of the fractions that give the probabilities. c. No. The probability of green is. There would have to be green marbles in the bag. There can be any multiple of marbles in the bag a. b. = 7. Game : possible, equally likely, fair Game : possible, unlikely, unfair Game : possible, equally likely, fair Game : possible, unlikely, unfair Game 5: not possible, impossible, unfair. a. b. 66 c. Yes; students might make a tree to show all the possibilities:,,,, 5, 6,,,,, 5, 6,, and. d. Karen wins on,,,,,, 5, and 6. Mia wins on, 6,,, 6,, 5, 56, and 6. So Mia has a greater chance of winning than Karen. Students can do this experimentally, but the theoretical probabilities are so close that it would be hard for them to design an experiment with a large enough number of trials to arrive at the correct conclusion. Extensions. Answers will vary. If students conduct enough trials, the two types of probability should be close. 5. Contestants should think twice before playing this game. A contestant wins by choosing the combination RR, YY, or BB, so P(match) =, or, and P(no match) = =. 9 Students may argue that having a in chance of winning $5,000 is worth the risk of losing the prizes won so far. 6. a. Bag Bag Outcome How Likely Is It?
5 b. c. 9 d. No; Jason didn t list all the outcomes for Bag. He only has two outcomes, which are and not-, and these are not equally-likely outcomes. He should have, and under Bag. For each outcome starting under Bag and going to Bag, he should have three branches, one for, one for, and one for. (Note: Make sure that students notice that the outcomes for each action in the diagram, in this case choosing from Bag and choosing from Bag, must have all the possible outcomes accounted for.) 7. a. HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, TTHH, THTH, TTTH, TTHT, THTT, HTTT, TTTT b. c. Answers will vary. 6 Possible Answers to Mathematical Reflections. If you analyze a situation by thinking about the range of possible equally likely outcomes, you can list all the possible outcomes and find the theoretical probability of an event. For example, to find the theoretical probability of choosing a block from a bag of blocks you would compute: number of blocks P() = total number of blocks This is called the theoretical probability because it is based on analyzing a situation to determine what should happen in theory.. a. Not necessarily; their experiments will probably give them different results, but their results should be close, especially if they each conducted a large number of trials. b. Yes; the theoretical probability in a given situation is the same no matter who finds it. c. They probably won t get the same results but their results will probably be close, especially if the experimental probability is based on a large number of trials. ACE ANSWERS Investigation Experimental and Theoretical Probability 57
Applications. a. P(green) =, P(yellow) = 2, or 2, P(red) = 2 ; three of the four blocks are not red. d. 2. a. P(green) = 2 25, P(purple) = 6 25, P(orange) = 2 25, P(yellow) = 5 25, or 5 2 6 2 5 25 25 25
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