What's $P(\{k\})$ in this exercise? The question is based on the following problem in the book Probability Essentials by Jacod:
 
Here is my question:

What does $P(\{k\})$ mean in the second problem?

The probability on a finite or countable measurable space $(\Omega,\Sigma)$ is determined by $P(\{\omega\})$ where $\omega\in\Omega$. As I understand, for the binomial distribution $B(p,n)$ the sample space is $\Omega=\{(a_1, a_2):a_1,a_2=0,1\}^n$. How does $P(\{k\})$ come out here?

$k$ is the number of success which is an event $\in\Sigma$. Namely trying $n$ times getting the success of $k$ times. Therefore this event has some probability $P(\{k\})$. I also think that it is defined in $4.1$ and there is an abuse of notation.
