Math 2.0

definition

Event

Definition

Let Ω\Omega be a sample space and let F\mathcal{F} be a σ-algebra on Ω\Omega. An event is an element of F\mathcal{F}, that is, a subset A⊆ΩA \subseteq \Omega with A∈FA \in \mathcal{F}. The event AA occurs in a trial when the outcome ω\omega of the trial satisfies ω∈A\omega \in A.

Remarks

When Ω\Omega is finite or countable, one usually takes F\mathcal{F} to be the power set of Ω\Omega. Then each subset of Ω\Omega is an event. For uncountable Ω\Omega, such as Ω=R\Omega = \mathbb{R}, one cannot in general give a probability to each subset. So only the sets in F\mathcal{F} are events. The set operations have probabilistic names: A∪BA \cup B is "AA or BB", A∩BA \cap B is "AA and BB", and Ω∖A\Omega \setminus A is "not AA". The empty set is the impossible event, and Ω\Omega is the sure event. Two events AA and BB are mutually exclusive when A∩B=∅A \cap B = \varnothing.

Explanations

L1 human

An event is a statement about the result of the experiment that is either true or false after the experiment. For a die, "the result is at least 55" is the event {5,6}\{5, 6\}. It happens when you roll a 55 or a 66.

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  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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Sources

  1. Web https://en.wikipedia.org/wiki/Event_(probability_theory), Events in probability spaces

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