硬币落在 1 英寸方格内：中奖概率

Coin in Square

专题: Strategy / 策略
难度: L4
来源: QuantQuestion

题目详情

In a common carnival game a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1 - inch squares. If the penny ( $\frac{3}{4}$ inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his penny back; otherwise he loses his penny. If the penny lands on the table, what is his chance to win?

解析

硬币直径 $3/4$ 英寸，半径 $r=3/8$ 。

要完全落在某个 $1\times 1$ 方格内，硬币中心必须距离任一边至少 $r$ ，因此中心可落区域为边长

1-2r=1-\frac{3}{4}=\frac{1}{4}

的内方格。

在“中心在方格内近似均匀”假设下，中奖概率等于面积比：

\boxed{\left(\frac{1}{4}\right)^2=\frac{1}{16}}.

（若考虑硬币可能落出桌面、或线条有宽度，中奖概率会更小。）