# Math Help - Probability question

1. ## Probability question

Show that for any $n$ events $A_1 ,A_2 ,...,A_n$ in a finite probability space, we have $P(A_1 \cup A_2 \cup \cdot \cdot \cdot \cup A_n) \leq P(A_1) + ...+P(A_n)$

2. Originally Posted by qwesl
Show that for any $n$ events $A_1 ,A_2 ,...,A_n$ in a finite probability space, we have $P(A_1 \cup A_2 \cup \cdot \cdot \cdot \cup A_n) \leq P(A_1) + ...+P(A_n)$
Have you ever heard of Boole's inequality?

http://www.andrew.cmu.edu/course/21-228/lec7.pdf

3. I really know nothing about probability and so this is more of a question then an answer. Doesn't this follow since the probability measure is by definition countably additive?

4. Yes but how is this done using only the following axioms (first principles)
1) $P(\emptyset)=0$
2) $P(\Omega)=1$
3) $P(A_1 \cup \cdot \cdot \cdot \cup A_n)=P(A_1) + ...+ P(A_n)$ for $n$ disjoint sets $A_1 , ...,A_n \subseteq \Omega$

5. Originally Posted by qwesl
Yes but how is this done using only the following axioms (first principles)
1) $P(\emptyset)=0$
2) $P(\Omega)=1$
3) $P(A_1 \cup \cdot \cdot \cdot \cup A_n)=P(A_1) + ...+ P(A_n)$ for $n$ disjoint sets $A_1 , ...,A_n \subseteq \Omega$
Use google, proofs this common are usually all over the internet.

http://www.math.ntu.edu.tw/~hchen/te...s/lecture2.pdf

If you don't like that one there are several others...

Originally Posted by Drexel28
I really know nothing about probability and so this is more of a question then an answer. Doesn't this follow since the probability measure is by definition countably additive?
Yes, I've seen a proof using this argument, wikipedias in fact.

6. Originally Posted by Anonymous1

Yes, I've seen a proof using this argument, wikipedias in fact.
Oops, probably should have wikid it first! God bless Wikipedia!