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Probability theory:

• Probability theory deals with predicting how likely it is that something will happen.
• If one tosses three coins, how likely is it that they will all come up heads?

Sample space (Ω)：

• In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment.
• A sample space is usually denoted using set notation, and the possible outcomes are listed as elements in the set.

Event:

• Let an event A be a subset of Ω.
• In the coin experiment, the first coin been a head, and the second and third coming down tail is one basic outcome, while any result of one head and two tails is an example of an event.

Source:

• Wikipedia.
• Foundations of statistical natural language processing.

A discrete probability function is any function  such that:

P(Ω)=1

Countable additivity: For disjoin sets )