What is the probability that the digit sum of a randomly chosen integer between 0000 and 9999 is divisible by 5? If I have a randomly selected integer between 0000 and 9999, what is the probability that the digit sum of that number is divisible by 5? 
[E.g. 1234 = 1 + 2 + 3 + 4 = 10]

I've started off with knowing that I have 2 options for the last integer, but I'm not sure where to go from there.
Hint: Pick the first three digits randomly first, and then focus on the last one.

It's similar to how the probability of getting the sum $7$ when throwing two dice can be seen to be $\frac16$ by noting that no matter what the first die shows, the result on the second die can make the sum $7$ in exactly one way.
Note: This answer is wrong, see in comments.

The sum of the four random digits will be between 0 and 36. There are seven numbers divisible by five in this range, so the probability is close to 20%, but not quite.