2022 AMC 8 Problem 25

Question

A cricket randomly hops between

4

leaves, on each turn hopping to one of the other

3

leaves with equal probability. After

4

hops, what is the probability that the cricket has returned to the leaf where it started?

\displaystyle \dfrac{2}{9}

\displaystyle \dfrac{19}{80}

\displaystyle \dfrac{20}{81}

\displaystyle \dfrac{1}{4}

\displaystyle \dfrac{7}{27}

Track only whether the cricket is on the starting leaf

If the cricket is not at the start, the next hop returns with probability

\frac13

Let

p_n

be the probability that the cricket is on its starting leaf after

n

hops. We have

p_0=1

.

If the cricket is on the starting leaf, the next hop must leave it. If the cricket is not on the starting leaf, exactly one of the

3

possible hops returns to the start. Therefore

p_{n+1}=\frac{1-p_n}{3}.

Thus

\begin{gathered} p_1=0,\quad p_2=\frac13,\quad p_3=\frac29,\quad \\ p_4=\frac{1-\frac29}{3}=\frac{7}{27}. \end{gathered}

Thus, the correct answer is E.

Problem 25 in other years