Let me build this up a bit.
There's five people on my bowling team, including myself. None of us are especially good at bowling; our averages range from about 120 to about 155 (Mine is the 155, which I think figures into this issue).
Many seasons ago, we decided to play a simple card game while bowling. For each bowling game (there are three games on a league night) we would put $1 on the table. When one of us got a spare or a strike, we would pick a card from a face-down, freshly shuffled and cut deck. At the end of the game, the player with the highest ranking poker hand won the "pot". There are also two wildcard jokers, and we agreed as a rule that a wildcard without an accompanying non-wild card was the least valued card in the deck... in order for it to be worth something, you needed to have something other than wildcards.
My teammates, from the beginning, just took the top card from the deck, flipped it over, and showed their hand. I had a slightly higher sense of the dramatic, so I always drew from the middle of the deck, and kept my cards hidden until the end. I have never drawn from the top or the bottom, and have no intention to ever do so.
Lately, several teammates have noticed I win the pot more often; anecdotally speaking this seems true although we've never tried to keep metrics on it.
I do have the highest average on our team; statistically, this would imply that I generally get more spares and strikes, and therefore have more cards from which I can create a superior poker hand.
However, tonight, I was told that what I was doing (pulling random cards from the middle of the deck instead of from the top) was cheating, as it somehow gave me better odds of getting a superior hand. I told them they were bad at math and it made absolutely no difference since none of us had any inkling of the order of the cards.
I do not see how it is possible for pulling from the middle of the deck to change the odds of a better poker hand, as the cards are well-shuffled, cut, and unknown for the duration of the game, but I concede that it is possible the one bad at math is me; as I mentioned earlier, it does seem I win more often than my teammates by a noticeable amount.
So the question is, does pulling from the middle of the deck when others are pulling from the top alter the probability of drawing a better hand in any mathematical manner?