OK, so I have another answer:
\mathbb{P}((\bigcap_{i=1}^{r}B_i )\cap (\bigcap_{i=r+1}^{n}B_i ))
=\mathbb{P}((\bigcap_{i=1}^{r}A_i )\cap (\bigcap_{i=r+1}^{n}A_i^c ))
=\mathbb{P}((\bigcap_{i=1}^{r}A_i )\cap ((\bigcup_{i=r+1}^{n}A_i )^c))
Now assuming that \mathbb{P}(\bigcap_{i=1}^{r}A_i )>0...
OK, so I have an answer:
\mathbb{P}((\bigcap_{i=1}^{r}B_i )\cap (\bigcap_{i=r+1}^{n}B_i ))
=\mathbb{P}((\bigcap_{i=1}^{r}A_i )\cap (\bigcap_{i=r+1}^{n}A_i^c ))
=\mathbb{P}((\bigcap_{i=1}^{r}A_i )\cap ((\bigcap_{i=r+1}^{n}A_i )^c))
Now assuming that \mathbb{P}(\bigcap_{i=1}^{r}A_i )>0 (if...
Homework Statement
Let (A_n : n\in \mathbb{N}) be a sequence of events in a probability space. Show that the events A_n are independent if and only if the \sigma-algebras \sigma(A_n)=\{\emptyset, A_n, A_n^c, \Omega\} are independent.
Homework Equations
For \sigma-algebras \mathcal{A}_i...