No two neighbouring lights on in a 3 by 3 grid

Nine light bulbs sit in a $3\times3$ grid. Independently, each bulb is on with probability $\tfrac12$. Two bulbs are neighbours if their cells share an edge. What is the probability that no pair of neighbouring bulbs are both lit?

$\tfrac{63}{512}$. We count on/off patterns with no two lit neighbours (independent sets of the $3\times3$ grid graph). Splitting on the centre bulb: with it on, only the corners are free ($16$ patterns); with it off, a case count gives $47$. That is $63$ of the $2^9=512$ equally likely patterns.