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# Counting question

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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?

Dec 11, 2021

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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.

Dec 11, 2021
edited by Melody  Dec 11, 2021