Five Great Activities Using Spinners. 1. In the circle, which cell will you most likely spin the most times? Try it.

Transcription

1 Five Great Activities Using Spinners 1. In the circle, which cell will you most likely spin the most times? Try it Marcy plays on her school basketball team. During a recent game, she was fouled and was sent to the free-throw line to shoot a one-and-one. Use a spinner to estimate the probability that she will score 0, 1, and 2 points if she has a free throw success rate of 60%. (note: one-and-one means that the player must make the first free throw to be eligible to attempt the second.) Construct a spinner with a 0.6 of the area shaded. Perform the experiment consisting of 25 one-and-one trials. Be sure to follow the rules of one-and-one free-throw shooting. Record the results of your experiment in the table on the next page.

2 Points Frequency Relative Frequency Based on the results of your experiment, how do you expect Marcy performed at the free-throw line? Why do you think this? Pool the data obtained by all the groups in your class to obtain a better estimate. In what ways is this model of free-throw shooting a realistic one? What are some of its shortcoming?

3 3. A breakfast- cereal company tries to increase its sales by putting small prizes in boxes of cereal, one prize in each box. If there are five kinds of prizes, estimate the number of boxes of cereal you would have to buy to obtain a complete set of prizes. Construct an appropriate spinner to simulate this situation. Assume that the various prizes occur in equal numbers and are uniformly distributed in boxes of cereal; that is, assume that the probability that a box of cereal contains certain animal is 1/5. Perform experiments that help you estimate a typical number of boxes to be opened to obtain a complete set of animals. Describe your experiment. How many boxes did you have to open to get a set? Pool the data obtained by all the groups in the class to obtain a better estimate. In what ways is this model of cereal box prizes a realistic one? What are some of its shortcomings?

4 4. For this experiment, you will need the spinners below. First, have one team member to use spinner P, another to use spinner R, and the third to record the results. The two players should simultaneously flick their spinners and the spinner landing on the higher number wins. Each pair of players should do 25 trials. After pooling the results from the class, which spinner wins most of the time? Spinner P Spinner R Spinner S Next, have the third member take spinner S, and play against the second player using spinner R. Spin 25 times. After pooling the results from the class, which spinner wins most of the time?

5 Finally, have the first member use spinner P play against the third member using spinner S. Spin 25 times. After pooling the results from the class, which spinner wins most of the time? Summarize your findings: Mathematical analysis: Calculate the probability of P beating R by making a table of equally likely outcomes. Spinner R Spinner P 20 P R R 60 P P R 70 P P R From the table, we see that spinner P is the winner 5 out of 9 times. Make similar tables for R and S, and for S and P and summarize the findings. Spinner S Spinner R Spinner P Spinner S Do you have any idea why the spinners are named P, R, and S?

6 5. A TV game show offers its contestant the opportunity to win a prize by choosing 1 of 3 doors. Behind 1 door is a valuable prize, but behind the other 2 doors is junk. After the contestant has chosen a door, the host opens 1 of the remaining doors, which has junk behind it, and asks if the contestant would like to stick with the initial choice or switch to the remaining unopened door. If you were the contestant, would you stick, switch, or possible choose the options at random.? You may be surprised to discover that your chance of winning vary depending on your strategy. Probability of Sticking: Suppose you always stick with your first choice. To simulate the sticking strategy, perform the following experiment using a spinner. Spin 30 times and tally your results. Determine the experimental probability of winning the prize if you always stick with your first choice. PRIZE JUNK JUNK Probability of Switching: No matter what the contestant chooses, the host will always open a door with junk behind it and ask if the contestant wants to switch. Perform the following experiment to simulate a switching strategy. Choose the first door randomly, using the spinner as before. With the switch strategy, you win if the pointer stops on junk but you lose if the pointer stops of prize. Explain why this happens. Then perform the experiment 30 times and compute the experimental probability of winning on a switch strategy. Probability of Randomly Sticking or Switching: Maybe the best strategy would be to choose a door and then decide on a stick or switch strategy. Choose your first door randomly, using the spinner as above. Then use the this spinner on the next page to decide whether you should stick or switch. What happens if your first spin selects junk and your second spin selects switch do you win the prize or not? Perform the experiment 30 times and compute the experimental probability of winning with this strategy.

7 STICK SWITCH The tree diagrams below represent the first stage of the game for the switch strategy and the stick strategy. Explain why we need only one stage of the tree diagram to determine whether we have won or lost. For each strategy, determine the theoretical probability of winning the prize. Junk (win) Junk (lose) Switch Junk (win) Stick Junk (lose) Prize (lose) Prize (win) The two-stage tree diagram represents picking a door at random and then deciding to stick or switch at random. Notice that if you randomly choose a junk door and then randomly spin switch, you switch choices when offered the opportunity and win the prize. Finish labeling the outcomes of the second stage and determine the theoretical probability of winning the prize with this stage. Switch prize Junk Stick Switch Random switch Junk Stick Switch Prize Stick