I'm reading my book on probability and it explains the 3rd Axiom as follows:
For any sequence of mutually exclusive events $E_1, E_2, …$ (that is, events or which $E_iE_j = \emptyset$ when $ i \ne j$ ):
$ P(\ \bigcup_{i=1}^\infty E_i\ )= \sum_{i=1}^\infty P(E_i) $
I know this is an "axiom" which is something assumed to be true. But usually there is a motivation or insight as to assuming WHY something is the case. Does anyone have any insights as to why someone would define this axiom or give a motivation for such a case?
Thanks in advance.
Best Answer
The reason it is defined in this way, is that Probability spaces are actually measure spaces, and the probability of an event is actually the measure of a set. So if you want to seek WHERE this definition comes from you should study first measure theory.
However, if you want to see why this applies consider a very simple example:
Take a fair die and toss it one time. Then the probability that each side appears is $1/6$. So, if you want the probability of the event $E=E_1\cup E_2\cup E_3$ , where $E_i$ is the event that number $i$ appears is : $$P(E)=P(E_1\cup E_2\cup E_3)=\sum\limits_{i=1}^3 E_i=1/6+1/6+1/6=1/2$$
It is obvious that ,at least, for a finite number of disjoint events it is natural to define the probability of the union as the sum of the probabilities.
Can you consider an example with infinite number of disjoint events?