Compare Winning Chances in a Card Game That Replays Ties

Two players take turns drawing from an ordinary 52-card deck. Player one draws a card uniformly at random, and player two then draws uniformly from the 51 cards left. Whoever has the higher rank wins. When the ranks are equal, both cards go back into the deck, it is shuffled, and the process repeats until someone wins. Is one player more likely to win than the other?

Constraints and Clarifications

The deck has 13 distinct ranks, each appearing four times. Suits are ignored for tie-breaking. After any tie, all 52 cards are returned, and the next round repeats the exact same random procedure.

Hint — Swap the two draws: For every ordered pair that gives player one a win, examine the pair formed by exchanging the two cards. Compare the probabilities of those two outcomes.

What a Strong Answer Covers

• A symmetry argument that still holds even though the cards are drawn without replacement.
• The chance of a tie in a single round, and why each individual round gives the first and second player equal win probabilities.
• Why repeating tied rounds does not favor either player.
• The final probabilities of winning the game, not just the first-round win probability.

Follow-up Questions

1. What is the expected number of rounds before a winner is decided?
2. How come player one having 52 possible cards versus player two's 51 does not automatically create an edge?

Overview: Use symmetry and the replay of ties to compare two players drawing without replacement, and work out the eventual win probabilities for the game.