Chapter 10Notes for InstructorsContentProbability is the focus of this chapter. Students should be able to:1. Understand and be able to explain the Law of Large Numbers.2. Understand the difference between empirical probability and theoretical probability.3. Calculate empirical and theoretical probabilities.4. Understand the following terms.(a) mutually exclusive events(b) independent events5. Apply the Addition Principle of Counting.6. Apply the Multiplication Principle of Counting.7. Evaluate permutations by applying the Multiplication Principle of Counting.8. Use permutations and the Multiplication Principle of Counting to evaluate combina-tions.9. Calculate odds.10. Use geometry to calculate theoretical probabilities involving spinners and dart boards.11. Construct experiments to simulate a real world situation.12. Make a probability treeYou will probably need about two weeks for this chapter. Certainly, it would be easy tospend a lot of time on this chapter, but I feel that it is unnecessarily heavy on the notation.I think the emphasis should be on the students’ ability to count, not their ability to decidewhether to use the combination formula or the permutation formula, since these formulaseasily fall from the Multiplication Counting Principle. I think you can teach this chapterwithout ever using the words “permutation” and “combination.” Looking ahead, you reallyneed to leave a lot of time for Chapters 11 and 12. In my experience, students have a lot ofdifficulty with the definitions in geometry and the unit concept in the measurement chapter.Manipulatives and TechnologyThere are a lot of manipulatives that can be used in this chapter including dice, spinners,and playing cards. I don’t think we have any of these in the Mathskellar, but some of thewooden blocks have been labeled as dice. You could label more if you need to do so. I1would suggest labeling them with two different colors, red and green, so that their samplespace matches the sample space in the book. Also, there are several problems which involvedrawing marbles out of a bag. This experiment can be mimicked with colored multi-linkcubes and a hat or bag.There are also several simulations in Winstats that could be helpful. The roll dice, thesample candy simulations will allow you to conduct some experiments without having diceand M&M’s rolling around the room. (I will say that actual M&M’s are very popular.) Youalso might find the spinner simulation and the dart-board simulation useful. In the spinnersimulation you can change the size of the sectors on the spinners by selecting Parameters...under the Edit menu. Similarly, you can change the size of the circle and the square in thedart-board.Notes and Suggestions:Notes on Section 10.1: Empirical Probability• Students should get a feel for the Law of Large Numbers by conducting an experimentrepeatedly and seeing the empirical probability change as the number of trials increases.• I used the M&M’srexperiment to illustrate the Law of Large Numbers. I don’t thinkit is possible to do the whole experiment in one class, but students should get the ideaeven if there isn’t time for each student to do the experiments 25 times. You couldmake them do the experiments outside of class. Then you would only need to combinethe data for the entire class and calculate the empirical probabilities during class. Ithink this would work well. You can find the actual color distribution for M&M’srathttp://www.mmmars.com/cai/mms/faq.html#what colors• I really like question 7, 14, 15, and 17–19 in Section 10.1.Notes on Section 10.2: Principles of Counting• You might want to quickly review the Multiplication Tree Model for multiplicationin Section 2.4 (p.115). This model combined with probability trees provides somejustification for the Multiplication Principle of Counting.• I think it is best to solve all of the problems in this section using only the Additionand Multiplication Principles of Counting. Permutations and combinations should flownaturally from the Multiplication Principle. (See the discussion on page 621.) I foundthat too many students were relying too heavily on the formulas for permutations andcombinations.Notes on Section 10.3:Theoretical Probability• The only new concepts here are odds and expected value, if you choose to cover them.2• On problem 10 in Section 10.3, your students will need to assume that the mark willindicate both a number and a letter. Therefore, the probability that the mark is onthe shaded area is310. If this is not the case, then we need to specify the radii of thecircles.WorksheetsI have included one worksheet with this