What is the probability of a SHA-512 hash containing both digits (0-9) and alphabets (a-f) all the time? Quite curious to know if ever a SHA-512 hash will contain only digits or only alphabets. Thanks in advance.
A cryptographic hash outputs a string of bits. 

Whether or not it appears as digits/alphabet letters is a matter of how the output is encoded and presented to the user. 

You could easily ensure that nothing but digits/alphabet letters in what is presented to a user by performing base conversion on the output, from a base-2 system (binary) using the symbols $0$, $1$ to a base-$n$ system using arbitrary symbols.

For example, If you encode the output in the hexadecimal number base, a user will see only hexadecimal values (0-9 + A-F).

If you wanted to operate under the assumption of a given number base, you would do some math to quantify the probability of a given subset of symbols occurring in the output - but you have to pick a specific encoding first before you can do so, as the choice of encoding will drastically chance the probability. 

For example, consider an encoding that uses (0-9 + A-Z + !) - The probability of sha512 outputting a string with only the symbols (0-9 + A-Z) is relatively high. While in the opposite case, if the encoding only uses (! + ?), the probability is non-existent.