1、 Review(Chapter 1 to Chapter 4)Jin QiVisualize Set Operations with Venn DiagramsDisjoint/Mutually Exclusive. It is said that two sets A and B are disjoint, or mutually exclusive, if A and B have no outcomes in common, that is, if A B = . The sets A1, . . . , An or the sets A1, A2, . . . are disjoint
2、 if for every i = j , we have that Ai and Aj are disjoint, that is, Ai Aj = for all i = j . The events in an arbitrary collection are disjoint if no two events in the collection have any outcomes in common.Intuition: In terms of events, A and B are disjoint if they cannot both occur.Properties of se
3、t operations- Commutative Laws- Associative Laws- Distributive Laws- Morgans LawsTheorem 1.4.11: Partitioning a SetFor every two sets A and B, and are disjoint and In addition, B and are disjoint, General Multiplication Rule: Suppose that an experiment has k parts (k=2), that the ith part of the exp
4、eriment can have possible outcomes (i=1,k), and that the number of possible outcomes in each part does not depend on what occurs in the other parts. Then the sample space S of the experiment will contain all vectors of the form , where is one of the possible outcomes of part i (i=1,k). The total num
5、ber of these vectors in S will be equal to the product . MULTIPLICATION RULEExample 1.7.1 - Routes between Cities: Suppose that there are three different routes from city A to city B and five different routes from city B to city C. MULTIPLICATION RULEThe number of total routes = 35 =15Rolling Three
6、Dice: The number of possible outcomes is 666 = 216Sampling without Replacement: Consider an experiment in which a card is selected and removed from a deck of n different cards, a second card is then selected and removed from the remaining n 1 cards, and finally a third card is selected from the rema
7、ining n 2 cards. A process of this kind is called sampling without replacement, since a card that is drawn is not replaced in the deck before the next card is selected. The total number of possible outcomes for all three selections is n(n 1)(n 2).PERMUTATIONSThe number of permutations of n elements
8、taken k at a time is:Permutations: Suppose that a set has n elements. Suppose that an experiment consists of selecting k of the elements one at a time without replacement. Let each outcome consist of the k elements in the order selected. Each such outcome is called a permutation of n elements taken
9、k at a time. We denote the number of distinct such permutations by the symbol PERMUTATIONSThe number of permutations of n elements taken k at a time is:Sampling with Replacement: Consider a box that contains n balls numbered 1, . . . , n. First, one ball is selected at random from the box and its nu
10、mber is noted. This ball is then put back in the box and another ball is selected (it is possible that the same ball will be selected again). As many balls as desired can be selected in this way. This process is called sampling with replacement.Example 1.7.11 - Obtaining Different Numbers: the probability of the event E that each of the k balls that are selected will have a different number
