Track / Stage 3 / 102 of 260 #102 of 1964

Problem 102

AMC 10/12, early questions

Combinatorics Difficulty 3.2 Multiple choice

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?

Pick one

Official solution

We will let the moves be complex numbers $a$ , $b$ , and $c$ , each of magnitude one. The 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 $1$ . (Can you show how?) Hence, the probability is $\boxed{\text{(C)} \frac {1}{4}}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.