Axiom of probability theory

Sample Space and Events

Axiom of probability theory

Sample space: A possible collection of all the results of an experiment.

1. For example, toss a coin, its sample space s={front, reverse}

2. If the experiment is to investigate the lifetime of a transistor (hours), then the sample space is the set of all real numbers greater than or equal to 0 s={x:0<=x}

It can be found that the sample space can be limited or infinite.

Event: Any subset of the sample space is called an event.

If the result of an experiment is contained in event E, then the event E is said to have occurred.

For example, in 1, another e={positive}, it means the event "toss a coin is positive"

Union: Any two events of the sample space S e,f,e and F is a new event, and his meaning is that if E and F have at least one occurrence, then E and f occur.

Intersection: Any two events of the sample space S E,f,e F is a new event whose meaning is that e-cross f occurs when and only if E and F are simultaneously sent.

Need to be carefully understood and this definition, union is the implication of the relationship, intersection is an equivalence relationship, in the Union, E and F occur, can not be introduced E and F at least one occurrence.

A number of events can be

Relationship between events: No intersection, there is an intersection (subset?). Equal? )

All definitions are described in a set language.

About orthogonal's law of De Morgan.

The naïve definition of event probability and its problems

The probability of an event is defined as the number of times the event occurred/the limit of the total number of times.

An event is a collection of sample space, independent of the experiment. The probability of an event is defined as the number of occurrences of an event in n times/n when the limit is greater than infinity.

Question 1: Whether the probability of any event is a real number, that is, whether he is convergent.

S is the sample space for an experiment, E is a subset of S, and P (e) is a real number

1) 0<=p (E) <=1

2) P (S) =1

3) If the event is incompatible, to satisfy the additive

At this point, we become P (e) as the probability of event E.

2016.02.25.21.22 Kee

Axiom of probability theory