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MATH / Beginning at point A in the diagram below, Dora selects one of the four possible directions with equal probability. Each time she comes to an intersection, she again randomly selec…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Beginning at point AA in the diagram below, Dora selects one of the four possible directions with equal probability. Each time she comes to an intersection, she again randomly selects one of the possible directions. What is the probability that, in her first four steps, she will walk completely around the gray square? Express your answer as a common fraction. [asy]size(100); fill((1,1)--(1,2)--(2,2)--(2,1)--cycle, gray(.6)); draw((0,0)--(0,3)--(3,3)--(3,0)--cycle, linewidth(1.5)); draw((0,1)--(3,1), linewidth(1.5)); draw((0,2)--(3,2), linewidth(1.5)); draw((1,0)--(1,3), linewidth(1.5)); draw((2,0)--(2,3), linewidth(1.5)); dot(MP("A", (1,2), NW)); [/asy]
Plain-text mathematical notation (without MathML)
Beginning at point A in the diagram below, Dora selects one of the four possible directions with equal probability. Each time she comes to an intersection, she again randomly selects one of the possible directions. What is the probability that, in her first four steps, she will walk completely around the gray square? Express your answer as a common fraction. [asy]size(100);
fill((1,1)--(1,2)--(2,2)--(2,1)--cycle, gray(.6));
draw((0,0)--(0,3)--(3,3)--(3,0)--cycle, linewidth(1.5));
draw((0,1)--(3,1), linewidth(1.5));
draw((0,2)--(3,2), linewidth(1.5));
draw((1,0)--(1,3), linewidth(1.5));
draw((2,0)--(2,3), linewidth(1.5));
dot(MP("A", (1,2), NW));
[/asy]
Original LaTeX notation
Beginning at point $A$ in the diagram below, Dora selects one of the four possible directions with equal probability. Each time she comes to an intersection, she again randomly selects one of the possible directions. What is the probability that, in her first four steps, she will walk completely around the gray square? Express your answer as a common fraction. [asy]size(100);
fill((1,1)--(1,2)--(2,2)--(2,1)--cycle, gray(.6));
draw((0,0)--(0,3)--(3,3)--(3,0)--cycle, linewidth(1.5));
draw((0,1)--(3,1), linewidth(1.5));
draw((0,2)--(3,2), linewidth(1.5));
draw((1,0)--(1,3), linewidth(1.5));
draw((2,0)--(2,3), linewidth(1.5));
dot(MP("A", (1,2), NW));
[/asy]

level

Level 4

type

Counting & Probability

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Source and history

task_id
7424fd82-4472-5e33-91d0-2f8dbfe6bc30
task_key
counting~5fand~5fprobability--test--7424fd82-4472-5e33-91d0-2f8dbfe6bc30
task_revision_id
1
config / split
counting_and_probability / test

Official source

initial import