you could help me solve this exercise I have looked in books for help but I still can not solve it. Thank you.
A load balancer in a cloud computing system is composed of N servers. The balancer, When assigning the connections to the servers, decide at random which one to use. But nevertheless, Due to some prioritization of the servers, 10% of them have a higher utilization level than the rest. Model the problem of access to the servers of the cloud computing system, for that purpose define clearly the sample space (Ω), the observations in the sample space (ω) and the probability function of the events of the problem. Also define an impossible event. Finally, define the random variables X and Y, which respectively represent the events of accessing a system server and accessing any server of the most used. Which of these random variables gives more information about the problem?
Let us call $S_1, \dots, S_N$ the servers. The random experiment is just the choice of the server so $\Omega = \{S_1, \dots S_N\}$, and the issues are all the $\omega = S_i$, $i \in \{1, \dots, N\}$. Now I will assume that $N$ can be divided by 10 (an assumption which is implicit since otherwise 10% of the servers won't make any sense), and say that the servers $S_1, \dots, S_{\frac{N}{10}}$ are the most frequently used. So we have $$ \mathbb{P}(S_i) = a \text{ for }i \in \{1, \dots, \frac{N}{10}\}, \quad \mathbb{P}(S_j) = b \text{ for} j \in \{\frac{N}{10} + 1, \dots, N \}, $$ and since the total probability must be one, $(a,b)$ satisfies $\frac{aN}{10} + \frac{9bN}{10} = 1$.
An impossible event would be for instance $A = $
The server$N+1$is chosenor $B=$Two servers are chosen at the same time.For the last question, in the current formulation I would say that $X$ gives almost no information (we always access a server), while $Y$ provides a real information on the problem (whether or not we have access one of the most used server).