Even outcome after shaving a pip off a random face

Start with an ordinary die showing $1$ to $6$. Choose one face uniformly at random and reduce its number by $1$. If this modified die is then rolled, what is the probability of an even result?

$1/2$. Lowering a face by one flips its parity. If an even face was changed (probability $\tfrac12$), two faces remain even; if an odd face was changed, four are even. So $\tfrac12\cdot\tfrac26+\tfrac12\cdot\tfrac46=\tfrac12$.