# Probability with a shrinking sample space

If I have a set that contains some lowercase and uppercase letters, for example $$S = \{a,b,c,d,e,f,g,h,V,W,X,Y,Z\}$$. Say I choose 4 letters from this set in a uniformly random way; however, each time I choose an element I remove it. I.e, each element may only be chosen once.

My sample space is initially $$S$$, after I choose 4 elements the size of my sample space would then be $$|S| - 4$$.

I'm not too sure about how I would go about calculating the probabilities of various things, for example consider the event $$B=\text{"The 4th letter chosen is uppercase"}$$, how would I calculate $$\Pr(B)$$?

If there's anything I can do to improve my question I'm happy to get feedback :)

It can be treated as a permutation counting problem where:

$$Pr(B) = \frac{\text{permutations with 4th letter UC}}{\text{permutations of 4 from 13}}$$

$$Pr(B) = \frac{5\cdot\frac{12!}{9!}}{\frac{13!}{9!}} = \frac{5}{13}$$