Difference between revisions of "2022 AMC 8 Problems/Problem 25"
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Revision as of 16:31, 28 January 2022
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
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 he’ll make it back. With this formula and the fact that 
we have: ![\[P_1 = \frac13, P_2 = \frac29, P_3 = \frac7{27},\]](http://latex.artofproblemsolving.com/d/e/6/de6c7b064858fe3fbd12c5d548aea7f0bc8c3d8d.png)
so our answer is 
~wamofan
See Also
| 2022 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Last Problem | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
