kindly answer the following
Fundamental rules of probability: 1. P(0) = 0, P(0) = 1 2. For any event A, P(A) 2 0. 3. For any countable list of disjoint events A1, A2, A3, . .. (i.e. AinA, = 0) for any i # j), P(UP, A;) = En, P(A;). Special case: for any finite list of disjoint events A1, A2, A3, . .. , An (i.e. Ain Aj = 0) for any i # j), P(U_, Ai) = Et_, P(Ai). Use only the fundamental rules of probability (and basic set operations) to prove Problems 1 and 2. Problem 2 (2 points) Prove that for any events A, B, P(A) = P(An B) + P(An Bc).(a) Prove or give a counter example: If An B - An C, then B = C. [b) Simplify AU(BU ((AU B) nC))" using basic set operations. Do not use Venn diagrams (except, perhaps, to check your result)! (c) Let A, B. C, and D be subsets of S. Using set operations (C, D.n, U,"), express the situation that exactly three of (A, B. C, D) occur. You can use Venn diagrams to help visualize the problem, but your answer must be expressed using set operations (e.g., An B).CSC 1351-03, Spring 2018, Lab 7 Name this project Set Create a project named Set that illustrates the use of generic classes and generic methods. Implementing a Generic Set Class DEFINITION 1. Generics is a way of defining classes, interfaces and methods by using type pa- rameters to represent types. A more technical term for generics is parametric polymorphism. A generic method, interface or class is fully specified at run time. In Java, a generic type can only be specialized using an object type. Every primitive type in Java has an associated object type (a wrapper class). DEFINITION 2. A set is a finite or Infinite collection of objects in which order has no significance, and multiplicity is generally also ignored. Members of a set are often referred to as elements and the notation a E A is used to denote that ar is an element of a set A. A set Is usually denoted as a list of elements. For example, {2, 3, 4,5, 6) is a set that contains five elements. In today's lab, you will Implement a generic class, the Set
class. The class will be generic but its methods will not be generic. Some methods will use the formal type parameter of the class. In this lab, we introduce a basic software engineering concept called composition - one object is composed of another. A set is composed of a list. We will use the Java API ArrayList, also a generic class, to Implement the Set class. Basic Set Operations DEFINITION 3. The cardinality of a set is the number of elements that the set contains. For example, the cardinality of A = {2,3, 4, 5, 6}, denoted |A| Is 5. DEFINITION 4. The intersection of two sets A and B is the set of elements common to A and B. This is written An B , and is pronounced "intersection" or "cap." DEFINITION 5. The union of two sets A and B is the set obtained by combining the members of each without allowing multiplicity. This is written A U B, and is pronounced "union" or "cup ." DEFINITION 6. The difference of sets A and B, denoted A - B, is the set of elements belonging to set A but not B.the patient is in the group treated with interferon alfa, and let B denote the event that the patient has a complete response. Determine the following probabilities. (a) P(A) (b) P(B) (c) P(AnB) (d) P(AUB) (e) P(A'UB) 2-81. A computer system uses passwords that contain exactly eight characters, and each character is one of 26 low- ercase letters (a-z) or 26 uppercase letters (A-Z) or 10 inte- gers (0-9). Let 2 denote the set of all possible passwords, and let A and B denote the events that consist of passwords with only letters or only integers, respectively. Suppose that all passwords in 2 are equally likely. Determine the probabil- ity of each of the following: (a) A (b) B (c) A password contains at least 1 integer. (d) A password contains exactly 2 integers. lying basic set operations to individual events. Unions of events, vents, such as AnB; and complements of events, such as A'-are bility of a joint event can often be determined from the probabili- it comprises. Basic set operations are also sometimes helpful in joint event. In this section, the focus is on unions of events. 2-1 lists the history of 940 wafers in a semiconductor manu- wafer is selected at random. Let / denote the event that the
