Problem 852

Coins in a Box

This game has a box of $N$ unfair coins and $N$ fair coins. Fair coins have probability $50 %$ of landing heads while unfair coins have probability $75 %$ of landing heads.

The player begins with a score of $0$ which may become negative during play.

At each round the player randomly picks a coin from the box and guesses its type: fair or unfair. Before guessing they may toss the coin any number of times; however, each toss subtracts $1$ from their score. The decision to stop tossing and make a guess can be made at any time. After guessing the player’s score is increased by $20$ if they are right and decreased by $50$ if they are wrong. Then the coin type is revealed to the player and the coin is discarded.

After $2 N$ rounds the box will be empty and the game is over. Let $S (N)$ be the expected score of the player at the end of the game assuming that they play optimally in order to maximize their expected score.

You are given $S (1) = 20.558591$ rounded to $6$ digits after the decimal point.

Find $S (50)$ . Give your answer rounded to $6$ digits after the decimal point.

盒中硬币

盒中有 $N$ 枚公平硬币和 $N$ 枚不公平硬币。公平硬币有 $50 %$ 的概率正面朝上，而不公平硬币有 $75 %$ 的概率正面朝上。

游戏开始时，玩家的分数为 $0$ ，在游戏过程中其分数可以为负。

每一轮，玩家从盒中随机选择一枚硬币，并猜测它是否公平。在猜测之前，玩家可以任意次抛掷这枚硬币，但是每次抛掷会扣 $1$ 分。玩家可以随时停止抛掷并给出猜测，猜对得 $20$ 分，猜错扣 $50$ 分。揭晓猜测结果后，这枚硬币会被移出游戏。

在 $2 N$ 轮之后，盒中将没有硬币，游戏结束。假设玩家总是采取最优策略以最大化期望分数，并记游戏结束时玩家的期望分数为 $S (N)$ 。

已知 $S (1) = 20.558591$ ，保留 $6$ 位小数。

求 $S (50)$ 并保留 $6$ 位小数。