Events and probabilities

1.1 Experiments with chance

1. experiment (or trial):a course of action whose consequence is not predetermined
2. probability space:
   • the set of all possible outcomes of the experiment
   • a list of all the events which may possibly occur as consequences of the experiment
   • an assessment of the likelihoods of these events.
3. probability theory:Given any experiment involving chance, there is a corresponding probability space, and the study of such spaces.

1.2 Outcomes and events

1. Sample Space:a list of all the possible outcomes of a particular experiment,The set of all such possible outcomes, we usually denote it by
2. elementary event:The Greek letter denotes a typical member of , and we call each member ω an elementary event.
3. Event Space:The collection of subsets of the sample space is called an event space if
   • is non-empty
   • if A F then
   • if then
4. an event space as being closed under the operations of taking complements and countable unions
5. An event space must contain the empty set and the whole set .
6. An event space is closed under the operation of finite unions
7. An event space is also closed under the operations of taking finite or countable intersections

1.3 Probabilities

1. Probability Measure:A mapping is called a probability measure on if
   • for ,
   • and ,
   • Countably Additivity:if , . . . are disjoint events in (in that whenever ) then

1.4 Probability spaces

1. Probability Space:A probability space is a triple of objects such that
   • is a non-empty set,
   • is an event space of subsets of ,
   • is a probability measure on .
2. Property:
   1. If , then
   2. If , then .
   3. If then.
   4. If then .
   5. If and then .

1.5 Discrete sample spaces

Let be an experiment with probability space. The structure of this space depends greatly on whether is a countable set (that is, a finite or countably infinite set) or an uncountable set. If is a countable set, we normally take to be the set of all subsets of , for the following reason. Suppose that and, for each , we are interested in whether or not this given is the actual outcome of ; then we require that each singleton set belongs to . Let . Then is countable (since is countable), and so may be expressed as the union of the countably many which belong to , giving that . The probability of the event is determined by the collection of probabilities as ranges over , since,

1.6 Conditional probabilities

1. (conditional) probability:If and , the (conditional) probability of given is denoted by and defined by
2. Theorem:If and then is a probability space where is defined by .

1.7 Independent events

1. Events and of a probability space are called independent if and dependent otherwise.
2. A family of events is called independent if, for all finite subsets of , The family is called pairwise independent if it holds whenever There are families of events which are pairwise independent but not independent

1.8 The partition theorem

1. A partition of is a collection of disjoint events with union.
2. Partition theorem:If is a partition of with for each , then This theorem has several other fancy names such as ‘the theorem of total probabilityand it is closely related to ‘Bayes’ theorem’.
3. Bayes’ theorem:Let a partition of the sample space with for each For any event with ,

1.9 Probability measures are continuous

1. A sequence of events in a probability space is called increasing if The union of such a sequence is called the limit of the sequence.
2. Continuity of probability measures:Let be a probability space. If is an increasing sequence of events in with limit , then