I draw thirty cards from a regular deck (4 suits x 13 cards each). The only fact I know is:
As I draw cards from the deck, the next draw is randomly chosen from the remaining cards in the deck
How can I find the probability that the last card drawn is a spade?
I have no doubt that it's $\large \frac{1}{4}$. But I don't know how to prove it formally.
In other words, how do I prove that the deck with the above property represents a fully random shuffle?