936 CHAPTER 13 Counting and Probability Number of combinations (p. 920) ( ) ( ) ( ) = = − C n r P n r r n n r r , , ! ! ! ! Number of permutations: Not distinct (p. 922) n n n n ! ! ! ! k 1 2 The number of permutations of n objects of which n1 are of one kind, n2 are of a second kind, . . . , and nk are of a kth kind, where = + + + n n n nk 1 2 Sample space (pp. 925–926) Set whose elements represent the possible outcomes that can occur as a result of an experiment Probability (p. 926) A nonnegative number assigned to each outcome of a sample space; the sum of all the probabilities of the outcomes equals 1. Probability for equally likely outcomes (p. 928) ( ) ( ) ( ) = P E n E n S The same probability is assigned to each outcome. Probability of the union of two events (p. 929) ( ) ( ) ( ) ( ) ∪ = + − ∩ PEF PE PF PEF Probability of the complement of an event (p. 930) ( ) ( ) = − P E P E 1 Objectives Section You should be able to ... Example(s) Review Exercises 13.1 1 Find all the subsets of a set (p. 911) 1 1 2 Count the number of elements in a set (p. 911) 2, 3 2–9 3 Solve counting problems using the Multiplication Principle (p. 913) 4, 5 12, 13, 17, 18 13.2 1 Solve counting problems using permutations involving n distinct objects (p. 917) 1–5 10, 14, 15, 19, 22(a) 2 Solve counting problems using combinations (p. 919) 6–9 11, 16, 21 3 Solve counting problems using permutations involving n nondistinct objects (p. 922) 10, 11 20 13.3 1 Construct probability models (p. 925) 2–4 22(b) 2 Compute probabilities of equally likely outcomes (p. 928) 5, 6 22(b), 23(a), 24, 25 3 Find probabilities of the union of two events (p. 929) 7, 8 26 4 Use the Complement Rule to find probabilities (p. 930) 9, 10 22(c), 23(b) Review Exercises 1. Write down all the subsets of the set { } Dave, Joanne, Erica . 2. If ( ) ( ) = = n A n B 8, 12, and ( ) ∩ = n A B 3, find ( ) ∪ n A B . 3. If ( ) ( ) = ∪ = n A n A B 12, 30, and ( ) ∩ = n A B 6, find ( ) n B . In Problems 4–9, use the information supplied in the figure. 4. How many are in A? 5. How many are in A or B? 6. How many are in A and C? 7. How many are not in B? 8. How many are in neither A nor C? 9. How many are in B but not in C? In Problems 10 and 11, compute the given expression. 10. ( ) P 8, 3 11. ( ) C 8, 3 6 4 5 20 20 C B U A 0 1 2
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