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  [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|$ and is the measurement operator, 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 $?