If I pick a random sequence of non-repeating 8 letters, what's the probability that the letters are in alphabetical order?

Question

For instance, abcdefgz works, but redecfed does not.

Answer 1

I assume that by "non-repeating", you mean "distinct". And by "random" I assume you mean that you're picking a sequence uniformly at random (i.e., all sequences are equally likely).

You can imagine decomposing the process of picking a sequence into two steps: first picking the set of $8$ letters, and then putting these $8$ letters into some order. There are $8!$ orderings (permutations) of these $8$ letters, and all of these are equally likely, and only one of these orderings has the letters in alphabetical order, so the probability that the letters are in alphabetical order is $$\frac{1}{8!} = \frac{1}{40320} \approx 0.000025$$

Answer 2

Whatever letters we pick, there are $8!$ equally likely orders they could appear in. So the probability they will be in the correct order is $\dfrac{1}{8!}$.