Question: > 
> Suppose you flip a fair coin three times,
>  find three events $A$, $B$, and $C$, such that no two of them are
>  independent, but $P(A \cap B \cap C) = P(A)P(B)P(C)$
> 
> 
> 

I am given the question above. From what I can tell, this is a trick question since:

$P(A \cap B \cap C) = P(A)P(B|A)P(C|B,A) = P(A)P(B)P(C)$

then 

$P(B|A)$ must equal $P(B)$ (and the same for $P(C|B,A) = P(C)$), which means 

$P(A \cap B) = P(A)P(B)$, that is, they are independent, which goes against the premise.

Am I missing something here or is this correct? (I feel like I'm missing something)

EDIT: If someone could answer this visually (using a venn diagram or something) I'd really appreciate it

Answer: 
> 
> Specifically, I would like to know, can I make a statement like: "All fields are rings, and all rings are groups"?
> 
> 
> 

That is all correct. A field satisfies all ring axioms plus some extra axioms, so a field is a ring. A ring is an Abelian group plus some more axioms, so each ring is a group.

A vector space is also an Abelian group with some extra axioms relating it to a field. The field is an indispensable part of the definition of the vector space.

If you define a vector space to be an Abelian group V which has multiplication defined with a field (or division ring ) $V\times F\to V$ satisfying some axioms, then you can replace F with another ring and do something similar, except that V is called a *module* over the ring rather than a vector space. In other words, rings and modules are a generalization of fields and vector spaces.
Every vector space is *over a field*. And every field is a vector space over itself.

So for instance $\Bbb R$ is a field. But we can also say that $\Bbb R$ is a vector space over the field $\Bbb R$. Likewise $\Bbb C$ can be considered a vector space over $\Bbb C$ (making it a one-dimensional vector space). Also $\Bbb C$ can be considered a vector space over $\Bbb R$ (making it a two-dimensional vector space).

If we use a ring instead of a field as our scalars, then we get a *module* rather than a vector space. Then every module is *over a ring*. And every ring is a module over itself.