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Some Basic Definitions � Probability Space: � Putting together the observations of the sections above, we have defined a probability space as follows: � A probability space is a triplet { Q , F, P } where: � Q is a nonempty set, called the sample space. � F is a collection of subsets - closed under countable set operations. The elements of F are called events. � P is a countable additive function from F into [0 , 1] such that P ( Q ) = 1, called a probability measure . � Example: � Books at SUT library ( sample space ) � CE books ( events ) � Probability of existence of a special book ( probability measure ) 94 Kasaei

Some Basic Definitions � Examples will clarify the probability space definition: � The main point is that one defines the probability of sets of outcomes (the events). � The probability should be countable additive (to be continuous). � Accordingly (to be able to write down this property), and also quite intuitively, the collection of events should be closed under countable set operations. 95 Kasaei

Sets & Set Operations � A set is a collection of objects , with each object in a set often referred to as an element or member of the set ( e.g ., the set of all image processing books). � The set with no elements is called the empty or null set , denoted by the symbol Ø. � The sample space is defined as the set containing all elements of interest in a given situation. 96 Kasaei

Basic Set Operations � The union of two sets A and B is denoted by: � The intersection of two sets A and B is denoted by: � Two sets having no elements in common are said to be disjoint or mutually exclusive , denoted by: 97 Kasaei

Basic Set Operations 98 Kasaei

Basic Set Operations ( Venn Diagram ) � It is often quite useful to represent sets and sets operations in a so-called Venn diagram , in which: � S is represented as a rectangle. � Sets are represented as areas (typically circles). � Points are associated with elements. � The following example shows various uses of Venn diagrams. � The shaded areas are the result (sets of points). 99 Kasaei

Basic Set Operations ( Venn Diagrams ) 100 Kasaei