AMC10 2020 A

AMC10 2020 A · Q13

AMC10 2020 A · Q13. It mainly tests Probability (basic), Coordinate geometry.

A frog sitting at the point (1, 2) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length 1, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices (0, 0), (0, 4), (4, 4), and (4, 0). What is the probability that the sequence of jumps ends on a vertical side of the square?

一只青蛙坐在点 (1, 2)，开始一系列跳跃，每跳平行于坐标轴，长度为 1，每次跳的方向（上、下、右、左）独立随机选择。序列在青蛙到达顶点为 (0, 0)、(0, 4)、(4, 4) 和 (4, 0) 的正方形边时结束。序列结束在正方形垂直边上的概率是多少？

(A) \frac{1}{2} \frac{1}{2}

(B) \frac{5}{8} \frac{5}{8}

(D) \frac{3}{4} \frac{3}{4}

(E) \frac{7}{8} \frac{7}{8}

Answer

Correct choice: (B)

正确答案：（B）

Solution

In the figure below, the solid circle is the starting point and the open circles are the possible ending points. After the first jump, the frog is at one of the points (0, 2), (1, 1), (1, 3), or (2, 2), each with equal probability. If it is at (0, 2), then the sequence of jumps ends, and the frog is on a vertical side of the square. In the other three cases, the frog is on a diagonal of the square, so by symmetry the sequence is equally likely to end on a horizontal side or a vertical side of the square. Thus the requested probability is $\frac{1}{4} \cdot 1 + \frac{3}{4} \cdot \frac{1}{2} = \frac{5}{8}$.

在下面的图中，实心圆是起点，空心圆是可能的终点。第一次跳后，青蛙以等概率到达 (0, 2)、(1, 1)、(1, 3) 或 (2, 2)。如果在 (0, 2)，则序列结束，青蛙在正方形的垂直边上。在其他三种情况下，青蛙在正方形的对角线上，因此由对称性，序列等可能结束在水平边或垂直边上。因此所求概率为 $\frac{1}{4} \cdot 1 + \frac{3}{4} \cdot \frac{1}{2} = \frac{5}{8}$。

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