## Checkpoint - Course 1, Unit 7 Simulation Models

In each unit, there is a final lesson and Checkpoint that helps students summarize the key ideas in the unit. The final Checkpoint will generally be discussed in class, with the teacher facilitating the summarizing, and students making notes in their Math Toolkits (teachers may just refer to this as "notes") of any points they need to remember, adding illustrative examples as needed. If your student is having difficulty with any investigation in this unit, this Checkpoint and the accompanying answers may help you recall the concepts involved, and give you the big picture of what the entire unit is about. If your student has completed the unit, then a version of this should be in his or her notes or toolkit. Students should also have Technology Tips in their toolkits, which may be useful for this unit.

Possible Responses to Unit Summary Checkpoint
a. Summarize the steps involved in using a simulation model to solve a problem involving chance.
• Identify all possible outcomes. Outcomes do not have to be equally likely, but the probability of each must be known, and any dependence between outcomes understood. These probabilities underlie the choice of a random device or how random numbers are assigned to outcomes.
• State your assumptions. For example, can you reasonably assume that the probabilities stated in the first step will not change from trial to trial? Are trials independent? Also, you must decide what would constitute a trial.
• Select a device that will produce random outcomes that model the problem. For example, if you have identified two outcomes A (defective product, say) and B (not defective), and one has a probability of 20% and the other has a probability of 80% then a coin or 6 sided die will not be appropriate, but a 10 sided die or random numbers 1-10 will work. Outcomes 1 and 2 on the die would be assigned to defective products. Outcomes 3-10 would be assigned to not defective products. You must say what your device is and what the outcomes mean. It is important that the device generate random outcomes, because they are being used to model random occurrences.
• Conduct one trial, recording the result in a frequency table, indicating what the result of the trial means in terms of the situation. The number of components in a trial depends on the assumptions you made earlier. For example, in the above situation, getting 1345548 in a trial would mean that only one defective occurred in 7 products. Be sure that your frequency table will actually answer the question asked.
• Repeat for a large number of trials, recording the results in the frequency table and a histogram. For example, this table shows the above trial recorded with a tally mark:

 Number of Defective Items per Week Frequency 0 1 / 2 3 4 5 6 7

• Summarize your results and give your conclusion.
b. Will a simulation give you an "exact" answer? Explain your reasoning.
The results of a simulation will not give an exact answer for the probability of an outcome. However, because of the Law of Large Numbers, the more trials one does, the more likely that the result will be close to the exact probability.
c. What does the Law of Large Numbers say about how many trials should be done in a simulation?
We can expect the estimated, experimental probability to approach the theoretical probability with more trials.
d. A letter to the Washington Post on May 11, 1993, suggested that China has more boys than girls because if the first child is a boy then the parents tend to stop having children. Based on your work in this unit, do you think this is likely to be the case? Write a response to the author of this letter explaining your reasoning.
It might seem intuitive that if parents are trying to have a male child, and will keep trying till they get a male child and then stop, that there would be more boys overall. However, there will be some large families with all girls, except the last child. To simulate this using a coin, let H = Boy, T = Girl, so you have 50% of getting a "boy." Recording 20 trials in a table gives, for example,

 Number of Boys in Family Frequency Number of Boys Born Number of Children Born 1 (H) 9 9 9 2 (TH) 4 4 (1 per family) 8 3 (TTH) 2 2 (1 per family) 6 4 (TTTH) 3 3 12 5 (TTTTH) 0 0 0 6 (TTTTTH) 2 2 12 etc. Total = 20 Total = 47

In this example, the proportion of boys was 20/47, or approximately 0.425 with only 20 trials. As you do more and more trials, the percentage of the population that would be boys will approach 50%. (Students explored this problem on pages 487-488 and should not have to repeat the simulation here.)

If you would like to see specific problems from Course 1, Unit 7, a link is provided to Examples of Tasks from Course 1, Unit 7. If you are interested in following up on the Statistics strand in general, then the Scope and Sequence will help you see where different concepts are introduced. On the Statistics page, you can read an explanation of the main statistics concepts as they are developed in all four courses.