Consider a card guessing game with a standard deck of 52 distinct cards (4 suits × 13 ranks).

1. The deck is shuffled, with all 52! permutations being equally likely.
2. Cards are revealed one at a time from the top until all 52 cards are shown.
3. Before each card is turned over, you must guess its exact identity (rank and suit).
4. You earn $1 for each correct guess. There is no penalty for incorrect guesses.
5. You can use any strategy and base each guess on all previously revealed cards.

Assume you are risk-neutral and want to maximize your expected total winnings.

Questions:

1. What is the maximum possible expected number of correct guesses (and hence expected winnings in dollars) under an optimal strategy?
2. Based on this expectation, approximately how much would you be willing to pay to play this game once (i.e., what is a reasonable approximate fair entry price)?

Explain your reasoning and any probabilistic arguments used.