I. Element of probabilistic models
1. Every probabilistic model involves an underlying process, called the experiment. ( Example. Flip two coins )
2. The experiment produces exactly one out of several possible outcomes. ( Example. four outcomes: {????, ????, ????, ????} )
3. The set of all possible outcomes is the sample space. ( Example. Ω = {????, ????, ????, ????} )
4. Event is a subset of sample space. (Example. ???? = {????, ????} , the event that the two coins give the same side.
5. The probability law assigns our knowledge or belief to an event ?? a number ??(??) ≥ 0. It specifies the likelihood of any outcome.
II. Probability Axioms
1. (Non-negativity) ??(??) ≥ 0, for every event ??.
2. (Additivity) For any two disjoint events ?? and ??, ??(?? ∪ ??) = ??(??) + ??(??) In general, if ??1, ??2, … are disjoint events, then ??(??1 ∪ ??2 ∪ ?) = ??(??1) + ??(??2) + ?
3. (Normalization) ??(Ω) = 1.
III. Discrete model & Continuous model
In discrete models, it holds that for any event ?? = {??1, … , ????}, ??(??) = ??(??1) + ? + ??(????). When the probability law is uniform, then ??(??) = |??| / |Ω|.
However, sample space can also be infinite, and continuous. For continuous sample spaces, the probabilities of the single-element events may not be sufficient to characterize the probability law.
A natural candidate: For a continuous model Ω = [0,1]. Define the probability on any subinterval [??, ??] ⊆ [0,1] to be ??([??, ??]) = ?? − ??. (i.e. Probability = “the length of the interval.”)
IV. Properties of Probability Laws
Consider a probability law, and let ??, ??, and ?? be events.
1. If ?? ⊆ ??, then ??(??) ≤ ??(??) .
2. ??(?? ∪ ??) = ??(??) + ??(??) − ??(?? ∩ ??) .
3. ??(?? ∪ ??) ≤ ??(??) + ??(??) .
4. ??(?? ∪ ?? ∪ ??) = ??(??) + ??(??‘ ∩ ??) + ??(??‘ ∩ ??‘ ∩ ??).
ps. ??‘ is the complement of ??.
To be continued: http://www.cse.cuhk.edu.hk/~syzhang/course/Prob15/ - Lecture notes - Week 3 - Page 16
<Math: Probability and Statistics>Probability basics
原文：http://www.cnblogs.com/mjust/p/5162725.html