2010 AMC 12B Problems/Problem 18

Problem

A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than $1$ meter from its starting position?

$\textbf{(A)}\ \dfrac{1}{6} \qquad \textbf{(B)}\ \dfrac{1}{5} \qquad \textbf{(C)}\ \dfrac{1}{4} \qquad \textbf{(D)}\ \dfrac{1}{3} \qquad \textbf{(E)}\ \dfrac{1}{2}$

Solution

Solution 1 (Complex Numbers)

We will let the moves be complex numbers $a$ , $b$ , and $c$ , each of magnitude one. The frog starts on the origin. It is relatively easy to show that exactly one element in the set $\{|a + b + c|, |a + b - c|, |a - b + c|, |a - b - c|\}$ has magnitude less than or equal to $1$ . Hence, the probability is $\boxed{\text{(C)} \frac {1}{4}}$ .

Solution 2

The first frog hop doesn't matter because no matter where the frog hops is lands on the border of the circle you want it to end in. The remaining places that the frog can jump to form a disk of radius 2 centered at the spot on which the frog first landed, and every point in the disk of radius 2 is equally likely to be reached in two jumps. The entirety of the circle you want the frog to land in is enclosed in this larger disk, so find the ratio of the two areas, which is $\boxed{\text{(C)} \frac {1}{4}}$ .

2010 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
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All AMC 12 Problems and Solutions

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2010 AMC 12B Problems/Problem 18

Contents

Problem

Solution

Solution 1 (Complex Numbers)

Solution 2

See also