Chef and the Combination Lock

Chef has a combination lock with $N$ wheels (numbered $1$ through $N$). For each valid $i$, on the $i$-th wheel, the integers from $0$ to $A_i$ (inclusive) are written in ascending order ($0$ and $A_i$ are also adjacent). Initially, one value on each wheel is selected uniformly randomly and independently. The lock operates in a very peculiar way: it opens when there is at least one wheel where