The odds do change when shuffling after returning a bad card. Say fifty cards, thirty bad. You draw a card, you have 60% chance of bad. (30/50). You get a bad card, add it back to the end of the deck, and next draw is essentially 29/49 chance of getting bad if you do not shuffle, because the player knows THAT bad card isn't next. If you shuffle, odds return to 30/50.
Draw a good card (20/50), don't return it, and the next draw is 30/49 bad whether shuffled or not.
So as previously stated, no need to shuffle when removing cards, only when returning them.



