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_______________________ - a type of diagram that helps you organize groups of data that have some items in common.A Venn Diagram228605207000Principle of Inclusion and Exclusion for 2 sets.If you are counting the total number of elements in 2 groups or sets that have common elements, you must subtract the common elements so that they are not included twice. nA∪B=nA+nB-n(A∩B)Example 3 There are 10 students on the volleyball team and 15 on the basketball team. When planning a field trip with both teams, the coach has to arrange transportation for a total of only 19 students.Use a Venn Diagram to illustrate this situation.Determine how many students are on both teams.Determine the number of students in the remaining regions of your diagram and explain what these regions represent.Principle of Inclusion and Exclusion for 3 SetsIf you are counting the total number of elements in 3 groups or sets that have common elements, you must subtract the common elements present in the intersection of 2 sets, and add the elements present in the intersection of all 3 sets.nA∪B∪C=nA+nB+nC-nA∩B-nB∩C– nA∩C+ n(A∩B∩C) Example 4 There are 900 employees at CantoCrafts Inc. Of these, 615 are female, 345 are under 35 years old, 482 are single, 295 are single females, 187 are singles under 35 years old, 190 are females under 35 years old, and 120 are single females under 35 years old. Use a Venn diagram to determine how many employees are married males who are at least 35 years old.Additive Counting Principle (or Rule of Sum):If one action can occur in m ways, a second in n ways, a third in p ways, and so on, and these actions cannot be performed together, then there are m + n + p...ways in which one of these actions can occur.To use the additive counting principle, two events must be mutually exclusive (events cannot be performed at the same time)Additive Counting Principle could be considered a special case of the principle of inclusion and exclusion that applies only when sets A and B have no elements in common, so that nA∩B=0Cases must be used to solve these problemsHow many ways can you draw a 7 or a Jack from a deck of cards?How many ways can you roll a 5 or a 12 with two dice?Jim is playing Junior Monopoly with his friends. It has one die. Jim, during his turn, either wants to roll a 1, or a number higher than 4. How many ways are there for Jim to roll either a 1, or a number higher than 4?Mutually Exclusive or Not?A single card is drawn from a deck. In how many ways could it be a heart or a face card?How many five card hands consists of either all clubs or all red cards?How many two digit numbers are odd or multiples of 5?How many ways are there to pick 2 girls or 2 boys from a group of four boys and five girls?How many ways are there to roll a 2 or an odd number with two dice? How many of the first 100 natural numbers are multiples of 2 or multiples of 5?Homework: pg. 229 #2, 3, 4 pg. 270 # 1-5, 7, 9 ................
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