In introductory undergraduate probability courses, even those with a focus on set theory, I've often seen the definition of a sigma field entirely skipped over. Indeed, I've often seen the definition of a probability space include the claim "where $\mathcal{F}$ is the set of all events". Now, I've got no doubt that sigma fields are necessary in probability theory, so I'm forced to ask. How is it that these undergraduate courses can manage to teach probability theory without making any mention of sigma fields? What do they lose by skipping over this definition? Or rather, what do they have to manipulate in order to avoid it?

At the very least, I can recall from my own time as a first year undergraduate that I would occasionally get confused over what exactly an "event" is and at least in principle, I can see where "the set of **all** events" part of the definition could be confusing. For example, rolling a six-sided dice would give you a perfectly valid sigma field of { {6}, {not 6}, $\emptyset $, $\Omega$ } which wouldn't seem to fit the earlier definition (e.g. where's the "not 5" event?), unless you cheat the sample space in some way that I would be unsure of the validity of.

Admittedly, there's an awfully cynical part of me that thinks that the answer may be "by hoping that nobody thinks too hard about it" and indeed, inspecting my notes from such courses suggests that may genuinely be the answer. But I'm hoping that the good people at Stack Exchange can show otherwise.