2022 AMC 8 Problems/Problem 25
Contents
Problem
A cricket randomly hops between leaves, on each turn hopping to one of the other
leaves with equal probability. After
hops what is the probability that the cricket has returned to the leaf where it started?
Solution 1 (Casework)
Let denote the leaf where the cricket starts and
denote one of the other
leaves. Note that:
- If the cricket is at
then the probability that it hops to
next is
- If the cricket is at
then the probability that it hops to
next is
- If the cricket is at
then the probability that it hops to
next is
We apply casework to the possible paths of the cricket:
The probability for this case is
The probability for this case is
Together, the probability that the cricket returns to is
~MRENTHUSIASM
Solution 2 (Recursion)
Denote to be the probability that the cricket would return back to the first point after
hops. Then, we get the recursive formula
because if the leaf is not on the target leaf, then there is a
probability that it will make it back.
With this formula and the fact that we have
so our answer is
.
~wamofan
Solution 3 (Also Casework)
We can label the leaves as shown:
See Also
2022 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
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