Let's consider this hypothetical game, in which there are 10 cards, and one of the cards is marked as a winning card. We put the cards in a stack and pass them around in a circle, each person choosing one card at random and passing the remaining stack on, until none are left.
What is the probability that I pick the winning card? Initially, for the first person choosing, the probability is 1/10. Does the probability increase with successive turns, because the chances that the winning card is picked is smaller at the beginning? Or is the probability the same 1/10 throughout the whole game? Is it more beneficial to go later than to go first? I tried to think about this through an expected value lens and got all confused.