Stress the key word for each set operation: “not” for complement; “and” for intersection; “or” for union. Section 7.2 Mention the order of set operations: If parentheses are present, simplify within them in the following order: 1) Take all complements. 2) Take the unions or intersections in the order they occur from left to right. If no parentheses are present, start with 1). To solve a survey problem, students must first be able to identify what type of object belongs to a certain region before they can determine how many objects belong to that region. Have students explore the union rule for sets by determining the number of cards that are red or a king in their decks. Compare this problem with the problem of determining how many cards are fives or sevens (disjoint sets).
xviii Hints for Teaching Finite Mathematics and Calculus with Applications Section 7.3 Students need to be able to identify the experiment, the number of trials, the sample space, and the event in each probability problem. Illustrate the basic probability principle using numerous examples with manipulatives. Section 7.4 Redo the examples used to explore the union rule for sets to explore the related union rule for probability. The complement rule is most useful for problems that contain statements of the form greater than, less than, etc. Section 7.5 Sometimes independent events can be thought of as events that have the same sample space. For example, when two cards are drawn one at a time with replacement, both draws have a sample space consisting of all 52 cards. If these cards are drawn, instead, without replacement, the sample space for the second card has been reduced to 51 cards. Emphasize that the notation P ( A | B ) reminds us how the sample space was reduced. Section 7.6 Point out that trying to calculate P ( F | E ) directly is sometimes impossible, too expensive, or too inconvenient. Thus, there is a need for Bayes’ theorem which allows for the indirect calculation of P ( F | E ) using P ( E | F ) . If a tree diagram is employed, then Bayes’ theorem can be stated as P ( F | E ) = the probability of the branch through F and E the sum of the probabilities of all branches ending in E . Point out that the branch in the numerator will also be one of the branches in the denominator. Section 8.1 To use the multiplication principle, break down the problem (the task) into parts. Draw a blank for each part. Fill in each blank with the number of ways that part of the task can be completed. Finally, multiply these numbers to obtain the solution. Permutations are a special case of the multiplication principle that does not allow for repetition. Section 8.2 An additional way to determine whether to use combinations or permutations is as follows: 1) Give a label to each of the n objects. 2) Pick r objects from the n objects.
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