Question

物体 $A$ 和 $B$ 在坐标平面中同时移动，每一步长度为一。物体 $A$ 从 $(0, 0)$ 出发，每步等概率向右或向上。物体 $B$ 从 $(5, 7)$ 出发，每步等概率向左或向下。下列哪一个数最接近两个物体相遇的概率？

Objects $A$ and $B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object $A$ starts at $(0, 0)$ and each of its steps is either right or up, both equally likely. Object $B$ starts at $(5, 7)$ and each of its steps is either left or down, both equally likely. Which of the following is closest to the probability that the objects meet?

$0.10$

$0.15$

$0.20$

$0.25$

$0.30$

答案：C

知识点：格路随机游走双射

难度评级：2010

小提示：

它们只能在各走 $6$ 步后、沿直线 $x+y=6$ 相遇。

They can meet only after each has taken $6$ steps, along the line $x+y=6$

大提示：

相遇路径对的数量等于连接两个起点的 $\binom{12}{5}$ 条单调路径，总路径对数为 $2^{12}$

The number of meeting path-pairs equals $\binom{12}{5}$ out of $2^{12}$ total pairs

解答：

两物体相距 $12$ 步，所以只能在每个物体各走 $6$ 步后，在反对角线 $x+y=6$ 上相遇。

把 $A$ 的六步路径与 $B$ 反向看的六步路径配对，相遇路径对与从 $(0,0)$ 到 $(5,7)$ 的单调路径一一对应，共 $\binom{12}{5}$ 条。

概率为 $\dfrac{\binom{12}{5}}{2^{12}}=\dfrac{792}{4096}\approx0.19$ ，最接近 $0.20$ 。

所以正确答案是 C。

The objects are $12$ steps apart, so they can only meet after each takes $6$ steps, on the anti-diagonal $x+y=6.$

Pairing $A$ ’s six-step path with $B$ ’s reversed six-step path matches meeting pairs one-to-one with the $\binom{12}{5}$ monotone walks from $(0,0)$ to $(5,7).$

The probability is $\dfrac{\binom{12}{5}}{2^{12}}=\dfrac{792}{4096}\approx0.19,$ which is closest to $0.20.$

Thus, the correct answer is C.

2003 AMC 12A 第 22 题

答案：C

难度评级：2010

小提示：

大提示：

解答：

其他年份的第 22 题