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.
(The diagram had some troubles being put here, so I will explain it.)
it is a box with three squares on each side, so the area would be nine units. The point A in on the top left square, and and the point is the intersection on the right bottome of it.
+36 Solution !: !: Their is a \(\frac{1}{4}\) chance the Dora the Explorer goes along the intended path. Since in order to go around the gray square, Dora needs to go 4 times in her intended path, so the chance that Dora goes along her intended path is \((\frac{1}{4})^4=\frac{1}{4^4}=\frac{1}{256}\). However, their are two ways to go around the gray square (clockwise and counterclockwise). So, the answer is \(2 \cdot \frac{1}{256} = \boxed{\frac{1}{128}}\).
Cphill, or whoever comes upon this post, can you check my work? I not as confident in this answer, its just - so straightforward, it's fishy.
P.S. Wondering how I know the graph? I got the same question on Alcumus
