Three ants are arranged on vertices of a triangle, one ant to a vertex. At some moment, all the ants begin crawiling along the sides of the triangel. Each one crawls along one of the two sides that connect to the vertex it is sitting on, with an equal probability of picking either.
Assuming that all the ants move with an equal speed, and that they keep crawling forever in the same direction along the triangle, what are the odds that no two will collide?
(In reply to answer
by K Sengupta)
Let us denote the three ants as A1, A2 and A3. Each of these ants will move either in the clockwise direction (P) or in an anticlockwise direction (N). Thus, the possibilities are:
(A1, A2, A3) = (N, N, N); (N, N, P); (N, P, N), (N, P, P), (P, N, N); (P, N, P); (P, P, N); (P. P, P)
Accordingly, we have precisely 8 equiprobable cases.
Now no two ants will ever collide if all the three ants move in the
same direction. Thus, the possibilities are:
(A1, A2, A3) = (N, N, N); (P, P, P)
Accordingly, only two of the events are favorable to a total of 8
Prob(Event that no two ants will ever collide)