A ladybug is walking at random on a hexagon. She starts at vertex A.Every minute, she moves to one of the two vertices (chosen at random) adjacent to the one she's currently on. What is the probability that, after 12 minutes, she is back at A?
+118608 She has 12 moves.
So that is 2^12 possible paths
Let her take r moves clockwise (positive) and 12-r moves anti-clockwise. (negative)
So she will end up at r - (12-r) = 2r-12 in a positive directions which is
2r-12 (mod6) = 2r (mod6)
2r(mod6) = 0 when r = 0,3,6,9,12
If r=0 then she goes anticlockwise every time so there is only one way she can do that 1
If r=12 then she goes clockwise every time so there is only one way she can do that 1
If r= 3 there are 12C3 = 220 possible paths
If r = 9 there are also 220 possible paths
If r=6 there are 12C6 = 924 possible paths
924+440+2 = 1366 possible paths where she will end up back at A
Probability that she will end up back at A
\(=\frac{1366}{2^{12}}\\ =\frac{683}{2^{11}}\\ \approx 0.335 \qquad\text{Correct to 3 decimal places}\)
** I am not claiming to be positive that this is completely correct.
