Lesson1 Waiting Times Monopoly is a board game that can be played by several players. Movement around the board is determined by rolling a pair of dice. Winning is based on a combination of chance and a sense for making smart real estate deals. While playing Monopoly, Anita draws the card shown below. She must go directly to the jail space on the board. Anita may get out of jail by rolling doubles with a pair of dice on one of her next three turns. Doubles means that both dice show the same number on the top. If she does not roll doubles on any of the three turns, Anita must pay a $50 fine to get out of jail. Anita takes her first turn and doesn t roll doubles. On her second turn, she doesn t roll doubles again. On her third and final try, Anita doesn t roll doubles yet again. She grudgingly pays the $50 to get out of jail. Anita is feeling very unlucky. 45 UNIT 7 PATTERNS IN CHANCE 45 UNIT 7 PATTERNS IN CHANCE
Lesson1 Waiting Times LESSON OVERVIEW One of the most common probabilistic situations is waiting for a specific event to occur: waiting for red to come up in roulette, waiting for doubles to appear in backgammon or Monopoly, waiting for a day of rain on days when the weather forecaster says there is a 20 percent chance of rain, etc. In this lesson, students will be introduced to the idea of a waiting-time distribution (also called the geometric distribution). They will construct frequency distributions and histograms of waiting-time distributions and discover that all have the same basic shape. In a standard waiting-time situation, the probability of a success on each trial is assumed to be the same, no matter what happened on previous trials. Students will explore situations in which this assumption is true and in which it is not. L Obj t To use simulation to construct frequency distributions for waiting-time situations when trials are independent, and to recognize the shape as skewed to the right To identify when trials are independent To recognize rare events To review finding the mean of a frequency distribution To review finding the probabilities of events associated with rolling a pair of dice LAUNCH full-class discussion You might want to begin this lesson by bringing in a Monopoly game and explaining the rules about how to get out of jail by rolling doubles. Present the situation on page 45 and let your students discuss it, using the Think About This Situation questions to facilitate the discussion. This will give you an opportunity to assess how much your students know about probability. LESSON 1 WAITING TIMES T45
Master 149 149 Think About This Situation Transparency Master LAUNCH continued Think About This Situation i Use with page 457. Anita s situation suggests several questions. How likely is it that a Monopoly player who is sent to jail (and doesn t have a Get Out of Jail Free card) will have to pay $50 to leave? As a class, think of as many ways to find the answer to this question as you can. In games and in real life, people are occasionally in the position of waiting for an event to happen. In some cases, the event becomes more and more likely to happen with each opportunity. In some cases, the event becomes less and less likely to happen with each opportunity. How does the chance of rolling doubles change each time Anita rolls the dice? On average, how many rolls do you think it takes to roll doubles? Do you think Anita should feel unlucky? Explain your reasoning. Master 150 150 Steps in a Simulation U N I T 7 P A TTERNS IN CHANCE 1. Be sure you understand the problem state it in your own words. 2. Identify all possible outcomes and determine the probability of each. 3. State the assumptions you are making. 4. Select a random device and describe how it models your problem. 5. Conduct one trial, recording the result in a frequency table.. Do a large number of trials, recording the results in your frequency table and a histogram. 7. Summarize your results and give your conclusion. Use with page 457. Master 151 151 Name Rolling Doubles Rolled doubles on first try 1 Rolled doubles on second try 2 Rolled doubles on third try 3 Rolled doubles on fourth try 4 Transparency Master U N I T 7 P A TTERNS IN CHANCE Date Activity Master Event Number of Rolls Frequency Total 100 See Teaching Master 149. a Have students record their conjectures to the how likely question so they can compare this with their result to Activity 1, Part d. You might also make a list of the ways students suggest to get the answer to this question. Some students surely will suggest trying the experiment of rolling dice. At this stage, students aren t expected to know how to do this problem theoretically, and many may not know that the probability of rolling doubles on one roll of a pair of dice is 1. b Although your students may know that Anita s chance of rolling doubles remains the same each time she rolls the dice (because that is what they have been told), many of them won t really believe it. If you are interested in the psychology of learning probability and want to test this, take a coin and try the following experiment with your students. If you don t discuss the correct answer at this time, you can come back to this scenario while discussing the final unit checkpoint. Students should be able to answer the questions correctly at that time. Ask, If I flip a coin ten times, how many heads do I expect to get, on the average? (Most students will answer five. Be sure that they understand that, on the average, half of the flips of a fair coin will be heads.) Flip the coin and announce that you got a head. Flip it again and announce that you got another head. This shouldn t seem strange to the class. Now ask them, When I have finished the ten flips, how many heads do you expect me to have? Almost every student will answer five. This answer indicates that the students expect the coin to balance out the first two flips of heads. In other words, they believe that tails are due and now have a probability greater than 1. There are eight 2 flips left and they expect only three of them to be heads. The correct answer is six. You have eight flips left and, with a fair coin, you expect four of them to be heads and four to be tails. With the first two flips of heads, that means you expect six heads for the ten flips. c Students may suggest trying an experiment or base their hunches on their game-playing experiences. They won t know that the theoretical answer is, but you don t need to tell them yet. (Students might want to look back at this question and see how the results from Activity 1 might help them decide whether or not Anita should feel unlucky.) Use with page 457. U N I T 7 P A TTERNS IN CHANCE EXPLORE INVESTIGATION 1 small-group investigation Waiting for Doubles See Teaching Master 150. By the end of this investigation, students will have constructed their first probability distribution, used it to give some estimates of probabilities, and calculated some theoretical answers for related questions. As you circulate among the groups, encourage them to describe what they are doing and to talk about the differences between finding experimental (Activity 2) and theoretical (Activity 3) probabilities. You also might remind them of the initial question, On average, how many rolls does it take to get doubles? See additional Teaching Notes on page T533C. T457 UNIT 7 PATTERNS IN CHANCE
Thi k Ab Thi Si i Anita s situation suggests several questions. a b c How likely is it that a Monopoly player who is sent to jail (and doesn t have a Get Out of Jail Free card) will have to pay $50 to leave? As a class, think of as many ways to find the answer to this question as you can. In games and in real life, people are occasionally in the position of waiting for an event to happen. In some cases, the event becomes more and more likely to happen with each opportunity. In some cases, the event becomes less and less likely to happen with each opportunity. Does the chance of rolling doubles change each time Anita rolls the dice? On average, how many rolls do you think it takes to roll doubles? Do you think Anita should feel unlucky? Explain your reasoning. INVESTIGATION 1 Waiting for Doubles In this investigation, you will explore several aspects of Anita s situation. For this investigation, we will change the rules of Monopoly so that a player must stay in jail until he or she rolls doubles. A player cannot pay $50 to get out of jail in this version of the game, and there is no Get Out of Jail Free card. 1. Now suppose you are playing Monopoly under this new rule and have just been sent to jail. Take your first turn and roll a pair of dice. Did you roll doubles and get out of jail? If so, stop. If not, roll again. Did you roll doubles and get out of jail on your second turn? If so, stop. If not, roll again. Did you roll doubles and get out of jail on your third turn? If so, stop. If not, keep rolling until you get doubles. a. Copy the frequency table below and put a tally mark in the frequency column next to the event that happened to you. Add rows as needed. Rolling Doubles Event Number of Rolls Frequency Rolled doubles on first try 1 Rolled doubles on second try 2 Rolled doubles on third try 3 Rolled doubles on fourth try 4 Total 100 LESSON 1 WAITING TIMES 457 LESSON 1 WAITING TIMES 457
b. With other members of your class, perform this experiment a total of 100 times. Record the results in your frequency table. c. Do the events in the frequency table appear to be equally likely? That is, does each of the events have the same chance of happening? d. Use your frequency table to estimate the probability that Anita will have to pay $50, or use a Get Out of Jail Free card, to get out of jail when playing a standard version of Monopoly. Compare this estimate with your original estimate in Part a of the Think About This Situation on page 457. e. Make a histogram of the data in your frequency table. Describe the shape of this histogram. f. Explain why the frequencies in your table are decreasing even though the probability of rolling doubles on each attempt does not change. 2. Later in this unit, you will analyze Anita s situation theoretically. That is, you will use mathematical principles to find the probability she has to pay $50. As a first step, in this activity you will explore how to find the probability of various events when two dice are rolled. a. Suppose a red die and a green die are rolled at the same time. Make a copy of the matrix-like chart below. R lli g T Number on Red Die 1 2 3 4 5 Di Number on Green Die 1 2 3 4 5 1, 1 3, 2 4, 5 What does the entry 3, 2 mean? Complete the chart, showing all possible outcomes when the two dice are rolled. How many outcomes are possible? Are these outcomes equally likely? Why or why not? 458 UNIT 7 PATTERNS IN CHANCE 458 UNIT 7 PATTERNS IN CHANCE
EXPLORE continued 1. c. The events in the table are not equally likely. The probability of rolling doubles for the first time on that roll decreases with each roll. This is reflected in the table through the decreasing frequencies. d. Probabilities will vary based on the results of the experiment. Using the sample frequency table from Part b, students will find that the estimated probability that Anita will get out of jail in the first three rolls is 17 14 12 or 0.43. Thus, the estimated 100 probability she will have to pay $50 is 1 0.43 or 0.57. (The theoretical probability is 0.58, as students will learn later.) Comparisons will depend on the students original estimate. e. Histograms will vary. A sample histogram follows; most histograms won t be this smooth, and they usually will continue on to the right. Rolling Doubles Masters 152a 152b Name 152a Rolling Two Dice Number on Red Die Use with page 458. 1 2 3 4 5 Date Number on Green Die 1 2 3 4 5 1, 1 3, 2 4, 5 Activity Master UNIT 7 PATTERNS IN CHANCE 20 Frequency 15 10 5 0 1 3 5 7 9 Number of Trials The histogram is skewed right. The bars decrease in height. Each bar is about 5 of the height of the bar to its left. f. There will be fewer people who will roll doubles for the first time on their second roll than who will roll doubles for the first time on their first roll. One way to picture this is to imagine all 100 people taking their first roll simultaneously. Those who get doubles on this roll (about 1 of the 100 people) leave the room. Everyone who remains then tries a second time to get doubles. About 1 will succeed. This isn t 1 of 100, but 1 of a smaller number. 2. See Teaching Masters 152a 152b. a. The 3, 2 entry means that when the dice are rolled, there is a 3 on the red die and a 2 on the green die. See additional Teaching Notes on page T533C. LESSON 1 WAITING TIMES T458
Master 153 153 Use with page 459. Checkpoint Transparency Master In this investigation, you explored the waiting time for rolling doubles. Suppose you compared your class s histogram of the waiting time for rolling doubles with another class s histogram. Explain why the histograms should or should not be exactly the same. What characteristics do you think the histograms will have in common? If two dice are rolled several times, what is the probability of getting doubles on the first roll? On the fourth roll? Be prepared to share your ideas with the entire class. U N I T 7 P A TTERNS IN CHANCE EXPLORE continued 2. b. Students should use the table they just constructed to compute these figures. 3 3 3 2 3 8 1 2 3 It may be helpful if you point out that it is not always the best policy to write fractions in the lowest terms when computing probabilities. For example, it s much easier to compare the probability of rolling a sum of 7 with that of rolling a sum of 11 if both probabilities have a denominator of 3. c. Yes, the probabilities are the same. The color simply helps organize the table of all possible outcomes. d. Each has a 1 chance of rolling doubles on her next turn. SHARE AND SUMMARIZE full-class discussion If your students are not sure of the response to Part b of the Checkpoint on page 459, you may wish to have them carry out the following experiment. The workload should be divided among groups of students. One group of students will represent Conchita trying to roll doubles on her first roll of the dice. That group will roll the dice many times, for example, 400 times, each time a first try, and count the number of doubles. This group now has an approximation of the probability of rolling doubles on the first try. The second group, representing Anita, would roll the dice until they did not get doubles twice in a row. The group would then try to get doubles on the third roll. Counting only the times the group tried a third roll, they should repeat this process 400 times (not doubles, not doubles, try to roll doubles). This group now has an approximation of the probability of rolling doubles on the third try, given that doubles didn t happen on the first or second try. The two groups probably won t get exactly the same probability. Before the experiment, discuss with the class how close the relative frequencies will have to be so that they believe the theoretical probabilities are the same and how far apart the relative frequencies will have to be so that they believe the theoretical probabilities are different. (With 400 repetitions, there is a 95 percent chance that the probabilities will be within 0.05 of each other.) One of the difficulties of simulation is the large number of trials needed to convince students that two probabilities are the same. If, for example, one student flips a nickel 100 times and another flips a penny 100 times, there is a very good chance that there will be a difference of 0.10 or more in the proportion of heads with the nickel and with the penny. One of the students could easily get 45 heads out of 100 and the other get 5 out of 100, for a difference of 0.11. Very roughly, if each experiment is repeated n times, there is a 95 percent chance that the two estimates will be no farther apart than 2. n See additional Teaching Notes on page T533D. T459 UNIT 7 PATTERNS IN CHANCE
b. If two dice are rolled, what is the probability of getting each of the following events? Doubles A sum of 7 A sum of 11 A sum of 7 or a sum of 11 Either a 2 on one or both dice or a sum of 2 c. Is the probability of rolling doubles the same if both dice are the same color? Explain your reasoning. d. Suppose that in playing the modified Monopoly game, Anita is still in jail after trying twice to roll doubles. Conchita has just been sent to jail. Does Anita or Conchita have a better chance of rolling doubles on her next turn? Compare your answer with that of other groups. Resolve any differences. Ch kp i In this investigation, you explored the waiting time for rolling doubles. a b Suppose you compared your class s histogram of the waiting time for rolling doubles with another class s histogram. Explain why the histograms should or should not be exactly the same. What characteristics do you think the histograms will have in common? If two dice are rolled several times, what is the probability of getting doubles on the first roll? On the fourth roll? Be prepared to share your ideas with the entire class. O Yo O Change the rules of Monopoly so that a player must flip a coin and get heads in order to get out of jail. a. Is it harder or easier to get out of jail with this new rule instead of by rolling doubles? Explain your reasoning. b. Play this version 24 times, either with a coin or by simulating the situation. Put your results in a table like the one in Activity 1 of Investigation 1. Then make a histogram of your results. c. What is your estimate of the probability that a player will get out of jail in three flips or fewer? LESSON 1 WAITING TIMES 459 LESSON 1 WAITING TIMES 459