Can independent events be represented as subsets of same sample space?

Question

I have an intuition that two events which are independent cannot be represented as subsets of same sample space as it would contradict the definition of independence! Is it true? Is there any argument for it?

Answer 1

Let $\Omega = \{(0,0),(0,1),(1,0),(1,1)\}$ be our sample space and let it be equipped with a uniform probability measure.

Let $A=\{(0,0),(0,1)\}$

Let $B=\{(0,0),(1,0)\}$

We have then $Pr(A)=\frac{1}{2}$, $Pr(B)=\frac{1}{2}$ and $Pr(A\cap B)=\frac{1}{4}=Pr(A)\cdot Pr(B)$

Both $A$ and $B$ are subsets of the sample space and since $Pr(A\cap B)=Pr(A)\cdot Pr(B)$ this implies that they are indeed independent events.

Remember the definition of independence is that $A$ and $B$ are independent events iff $Pr(A\cap B)=Pr(A)\cdot Pr(B)$ iff $Pr(A)=Pr(A\mid B)$ iff $Pr(B)=Pr(B\mid A)$

Answer 2

No, that's not true. On the contrary, for the whole set up to be meaningful, all events discussed within the same problem have to come from the same sample space.

What is the definition of independent events that you're thinking about? It's a certain formula that tells us something about the probabilities of those events. And roughly speaking it means that knowing that one has happened doesn't give us any information about the other one. Surely, in specific examples it sometime agrees with our intuition, but sometimes may seem counterintuitive. But that's a whole different story...