All sequences must end with {3rd 0, 1,1,1,1}
Clearly we must start with 0.
We can form
0001111
010101111
00101111
001101111
and that's it.
The probabilities of flipping these sequences is
\(\dfrac{1}{128}+\dfrac{1}{256}+\dfrac{2}{512}=\dfrac{1}{64}\)