Seeker's first spot turning up empty

Two players each choose one of five hiding places independently and uniformly at random (they may pick the same one). A seeker then searches one of the five places, also chosen uniformly at random. What is the probability that nobody is hiding in the place the seeker checks first?

$16/25$. Whatever spot the seeker picks, each hider independently avoids it with probability $\tfrac45$, so the spot is empty with probability $\left(\tfrac45\right)^2=\tfrac{16}{25}$.