Is an event a subspace of the sample space?
Is an event a subspace of the sample space?
In a lecture today, a professor of mine described an event as being "in" the sample space. When writing on the board, for a sample space $S$ and event $E$, it was denoted: $$E in S $$
This confused me, as I have always thought that events were subsets of the sample space, in which I case I would write: $$E subset S$$
When I asked after class, I was told that events are not subsets of the sample space. If they are not subsets of the sample space, then how are they defined?
For example, let $S$ be the 6 possible outcomes of rolling a 6-sided die. If we were interested in event $E$, where the number of pips is even, would $E$ not be a subset of all possible outcomes?
We are using standard definitions of events and sample spaces. A sample space is defined as all possible outcomes of an experiment, and an event is defined as some set of outcomes in the sample space.
– James Otto
Aug 25 at 1:07
And that answers your question definitively.
– whuber♦
Aug 25 at 13:16
1 Answer
1
When I asked after class, I was told that events are not subsets of
the sample space.
No you're correct. Events are subsets of the sample space. There could be a few sources of confusion, though.
An event $E$ is a subset of $S$, however it is an element of sigma-field or sigma-algebra generated by $S$. Perhaps he wrote something like $E in sigma(S)$. This is because the sigma-field is a set of sets.
I think I recall that certain textbooks differentiate between events and "simple events." In the case of your dice example, $1 in 1,2,3,4,5,6$, but $1 subset 1,2,3,4,5,6$. In the first case, a simple event is an element of the space, and in the second, it's a set. I don't know, though. I find this confusing myself.
I should have mentioned in my original post, I asked my professor after class and they were not referring to the sigma-algebra generated by $S$. I have never heard of "simple events", that is very interesting. However, the events we were discussing had more than one element, so I do not believe they could be classified as "simple events", and would need to be subsets of $S$.
– James Otto
Aug 25 at 1:00
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Unless your professor is using unusual definitions of "event" and "sample space," s/he is plainly wrong. (Even Wikipedia is unambiguously clear about that.) Perhaps you could quote the definitions they are using?
– whuber♦
Aug 23 at 13:14