The problem is not described with complete clarity. So some assumptions are needed in order to produce an answer. We assume that you are picking one of the $8$ locations independently and at random $3$ times, with all choices equally likely. In particular, it is assumed that repetition of location is allowed.
The probability of getting the bit sequence $0$, $1$, $0$ is then $\frac{5}{8}\cdot\frac{3}{8}\cdot\frac{5}{8}$, and the probability of getting the bit sequence $1$, $0$, $1$ is $\frac{3}{8}\cdot\frac{5}{8}\cdot\frac{3}{8}$. Add. Our probability is $ \frac{5}{8}\cdot\frac{3}{8}\cdot\frac{5}{8}+\frac{3}{8}\cdot\frac{5}{8}\cdot\frac{3}{8}.$ This simplifies to $\dfrac{15}{64}$.
Remark: The number $\dfrac{15}{64}$ is precisely your number $p_0p_1$. There is good structural reason for that. However, I think that the detailed analysis above is more informative, since it generalizes readily to other situations.
If repetition of location is not allowed, the analysis is quite similar, but the numbers change. We get $ \frac{5}{8}\cdot\frac{3}{7}\cdot\frac{4}{6}+\frac{3}{8}\cdot\frac{5}{7}\cdot\frac{2}{6}.$