# What is the difference between sample space and event space?

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?

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.
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.