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I am a little confused about the difference between sample space and event space. After reading some information, I want to take an example. If I am wrong, please correct me.

  • Sample space: all possible outcomes
  • Event: a subset of the sample space
  • Event space: all events

For a fair die:

  • Sample space: ${(1, 2, 3, 4, 5, 6)}$

  • Event: $(1)$ or $(2)$ or $(3)$ or $(4)$ or $(5)$ or $(6)$ or $(1,2)$ or $(1,3)$ or $(1,4)$ or $(1,5)$ or $(1,6)$ or $(2,3)$ or $(2,4)$ or $(2,5)$ or $(2,6)$ or $(3,4)$ or $(3,5)$ or $(3,6)$ or $(4,5)$ or $(4,6)$ or $(5,6)$ or $(1,2,3)$ or $(1,2,4)$ or $(1,2,5)$ or $(1,2,6)$,etc.

  • Event space: all events

Although the event is the subset of sample space, and event space is the all events. The sample space is actually not the same as event space, right?

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If you call the event space to be the space of all events, then in this case the event space here will be the power set of $\{1,2,3,4,5,6\}$ just as you mentioned. The relevant model assigns a probability equal to $\frac{\#\text{event}}{6}$ to an event. The event space being the power set of the sample space $\Omega$ will not be equal to $\Omega$.

In the measure theoretic foundations of probability theory, one does not generally take the set of events to be the power set of $\Omega$ because of measure theoretic considerations. In this case, the event space is not the power set but a smaller $\sigma$-algebra. Here, one cannot just define a probability for eacher $\omega\in \Omega$ but instead assigns a probability $P(E)$ to each event, that is, a measaurable subset of $\Omega$ with the relevant sigma algebra. However, in the case when $\Omega$ is at most countable, one can indeed take the $\sigma$-algbera to be the power set and get by with defining probabilities for singleton events.

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Everything you said is correct. If you want to write in a more mathematical way, you can consider a probability space $(\Omega,\mathcal{A},\mathbb{P})$, where $\Omega$ is a set, $\mathcal{A}$ is a $\sigma$-algebra of subsets of $\Omega$, and $\mathbb{P}:\mathcal{A}\to \mathbb{R}$ is a measure with $\mathbb{P}(\Omega)=1$. Then:

  • $\Omega$ is the sample space;
  • subsets of $\Omega$ are called events;
  • elements of $\mathcal{A}$ are called random events (those events who can be associated a probability).

If $\Omega$ is countable we usually take $\mathcal{A} = \wp(\Omega)$, and call that event space.

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