Track / Stage 3 / 231 of 260 #231 of 1964

Problem 231

AMC 10/12, early questions

Geometry Difficulty 3.8 Multiple choice

Sonya the frog chooses a point uniformly at random lying within the square
$[0, 6]$ $\times$ $[0, 6]$ in the coordinate plane and hops to that point. She then randomly
chooses a distance uniformly at random from $[0, 1]$ and a direction uniformly at
random from {north, south, east, west}. All of her choices are independent. She now
hops the distance in the chosen direction. What is the probability that she lands
outside the square?

Pick one

Official solution

WLOG, we assume Sonya jumps $0.5$ units every time, since that is her expected value.
If Sonya is within $0.5$ blocks of an edge, she can jump off the board. Let us examine the region that is at most $0.5$ blocks from exactly one edge.

Figure (Asymptote source)

import graph;  Label f; xaxis(0,6,Ticks(f, 6.0, 0.5)); yaxis(0,6,Ticks(f, 6.0, 0.5));  draw((0,0)--(6,0)--(6,6)--(0,6)--cycle); filldraw((0,0.5)--(0.5,0.5)--(0.5,5.5)--(0,5.5)--cycle,gray); filldraw((0.5,0)--(0.5,0.5)--(5.5,0.5)--(5.5,0)--cycle,gray); filldraw((6,0.5)--(5.5,0.5)--(5.5,5.5)--(6,5.5)--cycle,gray); filldraw((0.5,6)--(0.5,5.5)--(5.5,5.5)--(5.5,6)--cycle,gray);

If Sonya starts in this region, she has a

\dfrac14

chance of landing outside (there's exactly one direction she can hop to get out). The total area of this region is

4\cdot0.5\cdot5=10.

For this region, Sonya has a

\dfrac14

chance, so we multiply

10

\dfrac14

to get

2.5.

If Sonya is in one of the corner squares, she can go two directions to get out, so she has a

\dfrac24=\dfrac12

chance to get out. The total area is

0.5\cdot0.5\cdot4=1

, so this region yields

\dfrac12\cdot1=\dfrac12.

Adding the two, we get

3

. This is out of

36

square units of area, so our answer is thus

\boxed{\textbf{(B) }\tfrac{1}{12}}.

~Technodoggo

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.