Let's say I have a standard deck of 52 playing cards, and I need to find a card of a specific suit (lets say spades, for example) and the deck must be sufficiently randomized afterwards. One way to do this of course, is to actually pick up the deck, find a spade, and then shuffle up afterwards. Alternatively, though, what if I were to check the top card of the deck, and if it's a spade then remove it and assume the remainder of the deck is still just as sufficiently shuffled as it was before?
On its face, let's look at the apparent probabilities of drawing cards before and after removing the top card of a full deck this way: before - each card has a 1/4 chance to be any suit after - each card in the deck has a 12/51 chance to be a spade, and a 39/51 chance to be any other suit
This seems to me to work out, supposing you knew the probabilities were random upon the initial shuffling of the deck. However, I keep thinking back to the Monty Hall problem. I know it's not exactly applicable here, but I also know that it kind of shows how probabilities can be obscured and changed when you manipulate the objects in question.
What I'm looking for is this: If I find the top card of a shuffled deck to be the suit I'm in search of, do I need to shuffle again afterwards to ensure that my deck is still truly random? Or is the randomness of the remaining 51 cards preserved even after removing the top card?