MathDB
Cricket Hopping

Source: 2022 Winter AMC 8 #25

January 25, 2022
AMC 8

Problem Statement

A cricket randomly hops between 44 leaves, on each turn hopping to one of the other 33 leaves with equal probability. After 44 hops what is the probability that the cricket has returned to the leaf where it started?
<spanclass=′latex−bold′>(A)</span> 29<spanclass=′latex−bold′>(B)</span> 1980<spanclass=′latex−bold′>(C)</span> 2081<spanclass=′latex−bold′>(D)</span> 14<spanclass=′latex−bold′>(E)</span> 727<span class='latex-bold'>(A)</span>~\displaystyle\frac{2}{9}\qquad<span class='latex-bold'>(B)</span>~\displaystyle\frac{19}{80}\qquad<span class='latex-bold'>(C)</span>~\displaystyle\frac{20}{81}\qquad<span class='latex-bold'>(D)</span>~\displaystyle\frac{1}{4}\qquad<span class='latex-bold'>(E)</span>~\displaystyle\frac{7}{27}
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