What is the probability that the digit sum of a randomly chosen integer between 0000 and 9999 is divisible by 5?

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.

20%, or 1 in 5. I just counted them all in Excel. Consider that in each decade there will be 2 numbers divisible by 5. The first being 0000 and 0005. Then 0014 and 0019. And so on until 0091 and 0096. Then each century will be similar to the first except for a shift like we get with each decade, 0104 & 0109... 0190 & 0195. Likewise with the millennia. Consequently, the odds remain the same, 1 in 5.