jmc

For the ball to end up back in the center, the ball must deflect left 4 of the 8 times, and right the other 4 times. There are $\left(\genfrac{}{}{0ex}{}{8}{4}\right)$ ways to pick which 4 of the rows to deflect the ball left and then the other 4 to deflect it right. Each deflection is to the left with probability $\frac{1}{2}$ and to the right with probability $\frac{1}{2}$, so each possible path to the bottom occurs with probability ${\left(\frac{1}{2}\right)}^8$. Thus, the probability that 4 out of 8 deflections go left and the ball goes in the middle is $$\left(\genfrac{}{}{0ex}{}{8}{4}\right)\frac{1}{2^8}=\boxed{\frac{35}{128}}.$$