I am currently taking a course on quantum computing in college and some parts really confuse me. 

Starting from the basics, I understand that $ |\phi\rangle $  represents a state of a quantum system. I know that$ |\langle\psi|\phi\rangle|^2 $ gives a probability, but is it right to interpret it as the probability of the system in state $ |\phi\rangle $ collapsing to state $ \langle\psi| $?

We can perform operations on the system, and these are represented by operators, possible operations includes measurements, gates etc. It is represented by $ M|\phi\rangle $, and operating on the system leads to a change in the state of the system, $ M|\phi\rangle=|\psi\rangle $. However I came across this question:

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/bssXk.png
For a question like this,I would think of it as that we need to first apply V onto the state, and then do a measurement in the new basis, which would result in something like $ MV|\phi\rangle $ where $ M=\sum^{N-1}_i|\bar i\rangle \langle \bar i|$, but how do we go on to find the probability? Do I simply insert $ \langle \bar k| $ in front and turn it into $ |\langle \bar k|MV|\phi\rangle|^2 $?