AMC12 2023 B

AMC12 2023 B · Q20

AMC12 2023 B · Q20. It mainly tests Probability (basic), Circle theorems.

Cyrus the frog jumps $2$ units in a direction, then $2$ more in another direction. What is the probability that he lands less than $1$ unit away from his starting position?

青蛙Cyrus先向一个方向跳$2$个单位，再向另一个方向跳$2$个单位。他落在距起点不到$1$个单位处的概率是多少？

(A) \frac{1}{6} \frac{1}{6}

(B) \frac{1}{5} \frac{1}{5}

(D) \frac{\arctan \frac{1}{2}}{\pi} \frac{\arctan \frac{1}{2}}{\pi}

(E) \frac{2\arcsin \frac{1}{4}}{\pi} \frac{2\arcsin \frac{1}{4}}{\pi}

Answer

Correct choice: (E)

正确答案：（E）

Solution

Let Cyrus's starting position be $S$. WLOG, let the place Cyrus lands at for his first jump be $O$. From $O$, Cyrus can reach all the points on $\odot O$. The probability that Cyrus will land less than $1$ unit away from $S$ is $\frac{4 \alpha }{ 2 \pi}$. \[\sin \alpha = \frac{ \frac12 }{2} = \frac14, \quad \alpha = \arcsin \frac14\] Therefore, the answer is \[\frac{4 \arcsin \frac14 }{ 2 \pi} = \boxed{\textbf{(E) } \frac{2 \arcsin \frac{1}{4}}{\pi}}\]

设Cyrus的起点为$S$。不失一般性，设第一次跳后落在$O$点。从$O$点，Cyrus可到达圆$\odot O$上的所有点。落在距$S$不到$1$单位处的概率为$\frac{4 \alpha }{ 2 \pi}$。 \[\sin \alpha = \frac{ \frac12 }{2} = \frac14, \quad \alpha = \arcsin \frac14\] 因此，答案为 \[\frac{4 \arcsin \frac14 }{ 2 \pi} = \boxed{\textbf{(E) } \frac{2 \arcsin \frac{1}{4}}{\pi}}\]

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