If event A is a subset of event B, is it safe to say P(B-A) = P(B) - P(A)? In general, is it safe to say P(A+B) = P(A) + P(B), P(A*B) = P(A)*P(B), P(A/B) = P(A) / P(B), etc.?
Ok point taken. Anyways, here's a problem I'm kinda stumped on:
"Let $\displaystyle A_1$, $\displaystyle A_2$,...,$\displaystyle A_n$ be a sequence of increasing events of a sample space, i.e. $\displaystyle A_i$ is a subset of $\displaystyle A_{i+1}$, $\displaystyle i \geq 1$.
Let $\displaystyle B_1 = A_1$, $\displaystyle B_i = A_i - A_{i-1}$, $\displaystyle i \geq 2$. Note that $\displaystyle \bigcup_{i=1}^{\infty} A_i = \bigcup_{i=1}^{\infty} B_i$.
a. Use the events $\displaystyle B_1$, $\displaystyle B_2$,... to prove that $\displaystyle \lim_{n \rightarrow \infty} P(A_n) = P(\lim_{n \rightarrow \infty} A_n)$. This proves that probability is a continuous function, because the increasing sequence of events {$\displaystyle A_n$, $\displaystyle n \geq 1$} is a convergence sequence."
Does anyone have any idea how to prove lim P( ) = P[lim ( )] in general? I'll be happy if you can show me that; no need to help me completely for that whole part.