Finding a prize door with two hint spins

One of $3$ doors hides a prize of $\text{\char36}1$ million; the others are empty. Before choosing, you spin a wheel twice. On each spin, independently, the wheel reveals the prize door with probability $\tfrac35$ and otherwise reveals nothing. If neither spin reveals it, you pick one of the three doors uniformly at random. What is the probability that you end up with the prize?

$\tfrac{67}{75}$. Both spins are uninformative with probability $(\tfrac25)^2=\tfrac4{25}$; otherwise you know the door. So the success probability is $\tfrac{21}{25}+\tfrac4{25}\cdot\tfrac13=\tfrac{67}{75}$.