Compare Winning Chances in a Card Game That Replays Ties

Two players draw cards from a standard 52-card deck. The first player draws one card uniformly at random, then the second player draws one uniformly from the remaining 51 cards. The player with the higher rank wins. If both cards have the same rank, return both cards to the deck, shuffle, and repeat until a winner is determined. Does either player have a higher probability of winning? ### Constraints and Clarifications There are 13 ordered ranks with four cards of each rank. Suits do not break ties. After a tied round the full deck is restored, and the next round uses the same random experiment. ```hint Reverse the two draws For every ordered pair that makes the first player win, consider the pair obtained by exchanging the cards. Compare the probabilities of those two outcomes. ``` ### What a Strong Answer Covers - A symmetry argument that remains valid despite drawing without replacement. - The probability of a tied round and the equal per-round probabilities of a first-player or second-player win. - An explanation of why replaying ties does not give either player an advantage. - The eventual winning probabilities rather than only the probability of winning the first round. ### Follow-up Questions 1. What is the expected number of rounds until the game ends? 2. Why does the first player having 52 possible cards while the second has 51 not by itself imply an advantage?