Suppose I have a standard deck of cards, no cards missing and no abnormal cards added. If I shuffle the deck and draw one card at random, the odds of it being a spade are 1 in 4. That's pretty straightforward.

However, what if I put the card back in the deck, reshuffle, and draw another card? And I repeat this cycle X number of times (that is... X times I shuffle, X times I draw, and X-1 times I put the card back). I want to know what the odds are of me drawing a spade at least once.

And while we're at it, I'd also like to know the equation if there are a different number of suits. What if there's 3 suits and 39 cards? Or what if there's a 5th suit and 65 cards? Let's say that "1/s" represents the odds of me drawing a spade, with S representing the number of suits in the deck. And I draw after shuffling X number of times. What's the equation to determine the likelihood then?