This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: L A T E X Script Dieter Michael Schrottshammer Bernhard Ederer Dominik SchlagerWeidinger 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 ktuples. Example 2: If a test consists of 12 truefalse 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....
View
Full Document
 Fall '08
 Mendell
 Conditional Probability, Probability theory, Zagreb, Coin, Bernoulli trial

Click to edit the document details