Debug and optimize a card-drawing strategy

Problem Statement

You are given a deck of $N$ unique cards, each with an associated value. You are tasked with implementing a function select_optimal_hand(cards, k, strategy) that draws $k$ cards to maximize the sum of their values, given a specific, potentially flawed, strategy.

The Flawed Strategy

The current implementation iterates through the deck once. For each card, it calculates the probability of drawing a 'better' card in the remaining deck. If the probability is below a threshold $P$, it keeps the card; otherwise, it discards it.

Your task is to fix the function so that it correctly handles the state of the deck (which changes as cards are drawn) and performs efficiently for large decks.

Function Signature

def select_optimal_hand(cards: List[int], k: int, threshold: float) -> int:
    # cards: List of integers representing card values
    # k: Number of cards to draw
    # threshold: Probability threshold (0.0 to 1.0)
    pass

Requirements

1. Maintain a running state of the remaining deck.
2. Ensure the selection logic correctly accounts for the diminishing pool of cards after each draw.
3. Return the maximum possible sum if the strategy is followed perfectly until $k$ cards are selected.

Example

Input: cards = [10, 5, 8, 2, 9], k = 2, threshold = 0.5

• Card 10: 4 cards remaining, 0 better than 10. Prob = 0.0. Keep.
• Card 5: 3 cards remaining, 2 better than 5 (8, 9). Prob = 2/3 (0.66). Discard.
• Resulting logic should track state changes correctly.

Constraints

• $N$ up to 10,000.
• $k < N$.
• Time complexity should be better than $O(N^2)$.