A probability problem by Anthony Pham
An ant is on a regular tetrahedron, where all the edges are 1 meter long. The ant, starting on a vertex, randomly chooses and crawls to another vertex, and repeats. The probability that the ant is back to the same vertex after crawling 5 meters can be written as . If p and q are coprime, what is p + q ?
The answer is 101.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Shades of 2003 Aime II #13 in this one. Similar solution. On the first move the ant makes, P(A)=0 since it was already there and P(B)=1/3=P(C)=P(D).
From here, a recursive statement can be made that can be used to solve the problem. If on move "N",
P(A)=a
P(B)=b
P(C)=c
P(D)=d
then on move "N+1",
P(A) = (1/3 x b) + (1/3 x c) + (1/3 x d)
P(B) = (1/3 x a) + (1/3 x c) + (1/3 x d)
and so on...
after 5 moves P(A) = 20/81 so p+q=20+81=101
(sorry for ugly format)