1. ## Probability rules

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.?

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.?
no.

none of these are true in general. btw, what is P(A/B) supposed to mean?

for that matter, what is P(B - A) and P(A + B) supposed to mean? are you using them to mean P(B and notA) and P(A u B) respectively? no one uses that notation.

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.?
If event A is a subset of event B, then $P(A \cap B) = P(A)$.

by $P(B-A), \mbox{do you mean} P(B)-P(A \cap B)$?? Then your answer would be yes.

And what is P(A+B)?? Please clarify!

4. Ok point taken. Anyways, here's a problem I'm kinda stumped on:
"Let $A_1$, $A_2$,..., $A_n$ be a sequence of increasing events of a sample space, i.e. $A_i$ is a subset of $A_{i+1}$, $i \geq 1$.
Let $B_1 = A_1$, $B_i = A_i - A_{i-1}$, $i \geq 2$. Note that $\bigcup_{i=1}^{\infty} A_i = \bigcup_{i=1}^{\infty} B_i$.

a. Use the events $B_1$, $B_2$,... to prove that $\lim_{n \rightarrow \infty} P(A_n) = P(\lim_{n \rightarrow \infty} A_n)$. This proves that probability is a continuous function, because the increasing sequence of events { $A_n$, $n \geq 1$} 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.