notes and hw - L A T E X Script Dieter Michael...

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Unformatted text preview: L A T E X Script Dieter Michael Schrottshammer Bernhard Ederer Dominik Schlager-Weidinger 4. Oktober 2007 eg ) Tossing a coin twice. S= { HH,HT,TH,TT } A: Tail at the second toss. B: At least one head. A = { HT,TT } , B = { HH,HT,TH } A ∪ B = { HH,HT,TH,TT } = S A ∩ B = { HT } A = { HT,TH } , B = { TT } } ( A ∪ B ) = A ∩ B , ( A ∩ B ) = A ∪ B From the above example, ( A ∪ B ) = Ø = A ∩ B ( A ∩ B ) = { HH,TH,TT } = A ∪ B not a proof } A & B are mutually exclusive (disjoint): If A & B have no outcomes in com- mon. A B S eg ) Tossing a coin twice. A: Tail at the second toss. A = { HT,TT } D: Two heads. D = { HH } A & D are disjoint. 1 2 Counting } Tree Diagrams Example 1: Dinner. Soup: Clam chowder (CC), Broccoli (BR). Vegetable: French Fries (F), Salad (S). Meat: Chicken (C), Beef (B), Pork (P). P B C P B C P B C P B C F F S S CC BR 2 × 2 × 3 = 12 possible choices. } Product Rule Suppose a set consists of ordered collections of k elements. There are n , possible choices for the 1 st element. There are n 2 , possible choices for the 2 nd element. . . . . . . There are n k , possible choices for the k th element. ⇒ There are n 1 · n 2 · ... · n k possible k-tuples. Example 2: If a test consists of 12 true-false questions, in how many different ways can a student mark the test paper with one answer to each question? 2 · 2 · ... · 2 = 2 12 = 4096 . Definition: Factorial. n ! = n ( n- 1) · ( n- 2) · ... · 2 · 1 1! = 1 0! = 1 } Permutations: Order is important. 2 Definition: An ordered sequence of r objects from a set of n distinct objects: permutation of size r of the objects. n P r = n ( n- 1) · ( n- 2) · ... · ( n- r + 1) = n ! ( n- r )! Example 3: A committee consists of 10 members. #possible choices of the chairman and the vice chair: 10 P 2 = 10! (10- 2)! = 10! 8! = 10 · 9 = 90 } Combinations: Neglect the order....
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notes and hw - L A T E X Script Dieter Michael...

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