Recent content by Richard27182

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...

above answer does not work, I've applied De morgan's laws incorrectly

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...