Making Decisions With Probability
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- Barnaby Patrick
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1 Making Decisions With Probability! Spring vacation has arrived! Kalvin thinks he can stay up until 11:00 P.M. every night. His father thinks Kalvin will have more energy for his activities (such as roller blading, cleaning out the garage, or washing dishes) during his vacation if he goes to bed at P.M. 3.1 Designing a Spinner Getting Ready for Problem 3.1 Kalvin makes the three spinners shown below. Kalvin hopes that his father lets him use one of the spinners to determine his bedtime. Spinner 1 Spinner 2 Spinner 3 11:00 11:00 11:00 11:00 Which spinner gives Kalvin the best chance of going to bed at 11:00? Explain. Investigation 3 Making Decisions With Probability 39
2 Kalvin decides to design a spinner that lands on 11:00 the most. To convince his father to use this spinner, Kalvin puts three spaces, two 10:00 spaces, and one 11:00 space on the spinner. However, he uses the biggest space for 11:00. Kalvin hopes the pointer lands on that space the most. Which time do you think is most likely to occur? 10:00 11:00 10:00 Problem 3.1 Finding Probabilities With a Spinner A. 1. Find the experimental probability that the pointer lands on, on 10:00, and on 11: After how many spins did you decide to stop spinning? Why? 3. Suppose Kalvin spins the pointer 64 times. Based on your experiment, how many times can he expect the pointer to land on, on 10:00, and on 11:00? B. 1. What is the theoretical probability that the pointer lands on, on 10:00, and on 11:00? Explain. 2. Suppose Kalvin spins the pointer 64 times. Based on your theoretical probabilities, how many times can he expect the pointer to land on, on 10:00, and on 11:00? 3. How do your answers to Question A part (3) and Question B part (2) compare? C. Describe one way Kalvin s father can design a spinner so that Kalvin is most likely to go to bed at. 40 How Likely Is It?
3 3.2 Making Decisions Kalvin begins to think that probability is a good way to make decisions. One day at school, Kalvin s teacher, Ms. Miller, has to decide which student to send to the office to get an important message. Billie, Evo, and Carla volunteer. Kalvin suggests they design a quick experiment to choose the student fairly. Getting Ready for Problem 3.2 Which of these items can Kalvin s class use to choose a messenger? How can they make the decision fair? a coin a six-sided number cube colored cubes playing cards a spinner Investigation 3 Making Decisions With Probability 41
4 Problem 3.2 Analyzing Fairness Two suggestions for making a decision are shown in each question. Decide whether the suggestions are fair ways to make the decision. Explain your reasoning. A. At lunch, Kalvin and his friends discuss whether to play kickball, soccer, baseball, or dodgeball. Ethan and Ava each have a suggestion. Ethan: We can make a spinner that looks like this: Ava: We can roll a number cube. If it lands on 1, we play kickball. A roll of 2 means soccer, 3 means baseball, 4 means dodgeball, and we can roll again if it s 5 or 6. B. The group decides to play baseball. Tony and Meda are the team captains. Now they must decide who bats first. Tony: We can roll a number cube. If the number is a multiple of three, my team bats first. Otherwise, Meda s team bats first. Meda: Yes, let s roll a number cube, but my team bats first if the number is even and Tony s team bats first if it s odd. C. There are 60 sixth-grade students at Kalvin s school. The students need to choose someone to wear the mascot costume on field day. Huey: We can give everyone a number from 1 to 60. Then, we can roll 10 number cubes and add the results. The person whose number is equal to the sum wears the costume. Sal: That doesn t seem fair. Everyone should have a number from 0 to 59. In one bag, we can have blocks numbered 0 to 5. In another bag, we can have blocks numbered 0 to 9. We can select one block from the first bag to represent the tens digit and one block from the second bag to represent the ones digit. Kickball Soccer Baseball Dodgeball 42 How Likely Is It?
5 3.3 Scratching Spots Have you ever tried to win a contest? Probability can often help you figure out your chances of winning. Tawanda s Toys is having a contest. Any customer who spends at least $10 receives a scratch-off prize card. Each card has five gold spots that reveal the names of video games when you scratch them. Exactly two spots match on each card. A customer may scratch off only two spots on a card. If the spots match, the customer wins that video game. It can be difficult to get enough prize cards to conduct an experiment. So, you can design a related experiment to help you find the probability of each outcome. A model used to find experimental probabilities is a simulation. One way you can simulate the scratch-off card is by using five playing cards. First, make sure that exactly two out of the five cards match. Place the cards facedown on a table. While your eyes are closed, have a friend mix up the cards. Then open your eyes and choose two cards. If the cards match, you win. Otherwise, you lose. Can you think of another way to simulate the scratch-off cards? Problem 3.3 Using a Simulation A. Use the card simulation above to find the probability of winning. B. Examine the different ways you can scratch off two spots. Find the theoretical probability of winning with one prize card. C. Suppose you have 100 prize cards from Tawanda. 1. How many video games can you expect to win? 2. How much money do you need to get 100 cards? D. Tawanda thinks she may lose money with this promotion. The video games she gives away cost her $15 each. Will Tawanda lose money? Why or why not? Investigation 3 Making Decisions With Probability 43
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