Chapter3.1&3.2 lecture notes - Chapter 3.2 Counting Multiplication rule • Tree diagrams • Theorem 3.1 If sets A 1 A 2,….A k contain

AMS 310.01 Class notes September 2, 2003 Chapter 3.1, 3.2 Chapter 3.1 Sample Spaces and Events Basic definitions to be remembered: • Experiment -- Example: tossing two fair dice, one red and the other white) • Sample space is the set of all possible of outcomes of the experiment. • Sample spaces may be finite, discrete and countable, or continuous. • Event—subset of the sample space; Example : { the sum of spots on the red and white dice is equal to 7} • Events are combined by unions, intersections, and complements. Combinations of events are referred to as compound events. Algebra of set theory • Venn diagrams can be used to represent events and compound events. Example : Let {C}denote {ore contains copper} and let {U} denote {ore contains uranium} (1) Represent these two events with Venn diagrams (2) Identify four distinct events which have no intersection as follows: (1) C ∩ U (2) C ∪ U , (3) C ∩ U (4) C ∩ U, (3) Identify C ∪ U, U C ∪ , U C ∩ , C ∪ U , C and U in terms of 1-4 above and in words.

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Unformatted text preview: Chapter 3.2 Counting Multiplication rule • Tree diagrams • Theorem 3.1: If sets A 1 , A 2 ,….A k contain, respectively, n 1 , n 2 , …n k elements, there are , n 1 n 2 …n k-1 n k ways of choosing first an element of A 1 and then an element of A 2 ….. and finally an element of A k Example : 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? Permutations • n r P n n r =-! ( ) ! is the number of permutations of r objects selected from a set of n distinct objects. Example: In how many different ways can one make a first, second, third, and fourth choice among 12 firms leasing construction equipment? Combinations • The number of ways in which r objects can be selected from a set of n distinct objects is ( 29 r n n r n r =-! ! ( ) ! . Example: In how many ways can one choose four people for a committee from a total of 8 people? 1...
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