A frog, namely, Sally, is jumping about the vertices of a hexagon, \(ABCDEF\) , each time jumping to one of the adjacent vertices with equal probability. Let Sally start her daily workout in \(A\), and a mine be located in \(D\). Every second Sally must make her random jump (as described above). What is the probability that Sally will be alive after \(3133\) seconds, expressed as a decimal (and probably in scientific notation)?
As a bonus, can you generalise for \(n\) seconds?
This problem is a part of my froggy, soggy set.