Number of possible events is $2^N$ ( $N$ is the number of outcomes )? A probability space is a random process or experiment with
three components:
–

$Ω,$ the set of possible outcomes $O$

*number of possible outcomes = $|Ω| = N$

*$F$, the set of possible events $E$
      - an event comprises $0$ to
$N$ outcomes


number of possible events
= $| F | = 2^N$ 


Here I am not able to extract the exact meaning of number of possible outcomes. In case of tossing a coin, we have only $2$ outcomes. So, $N = 2$. So, either we have head or tail. According to the above definition, $|F| = 2^N = 2^2 = 4.$ How come this formula or axiom is valid in probability space ?
The answer by drhabs is wrong, his answer misses the null set. 2^n comes from the fact that for each element in your set you have 2 choices, to either include or not include it in a subset. For the example Ω={H,T} you will have the following subsets:


{H,T} (every set is a subset of it's self)
{H} 
{T} 
{} (null set)
Lets first see what an event means.
An event is not one outcome, but is a set.
For example in a problem of tossing a coin, the outcomes are $H$ &amp; $T.$ 
So the outcome set (Lets say $O$) , $O = \{H,T\}.$
Now an event is a subset of $O$, which means that it will contain the nullset, each single outcome and their combinations. i.e $E$ (all possible events) = $\{0,H,T,\{H,T\}\}.$

To understand why the number of events is equal to $2^N$ , lets take a outcome set $O$ having $N$ elements.
Null set can be selected in $1$ way
A event with one out the $N$ outcomes will be selected in $N$ ways,
with $2$ of the outcomes in NC2 ways and so on
when you add all of them,
its the binomial expansion of $2^N.$