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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?

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    $\begingroup$ 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? $\endgroup$ – whuber♦ Aug 23 '18 at 13:14
  • $\begingroup$ 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. $\endgroup$ – James Otto Aug 25 '18 at 1:07
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    $\begingroup$ And that answers your question definitively. $\endgroup$ – whuber♦ Aug 25 '18 at 13:16
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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.

  1. 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.

  2. 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.

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  • $\begingroup$ 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 $S$. $\endgroup$ – James Otto Aug 25 '18 at 1:00
  • $\begingroup$ @JamesOtto the dice example should work well to settle what kind of definitions are being used and whether or not your professor is talking about elementary events. Eventually these definitions are best made clear with practical examples to make them alive. $\endgroup$ – Sextus Empiricus Dec 21 '18 at 18:39

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