Let \( p_n \) be the probability that Gracie returns home after exactly \( n \) steps. We want to compute \( p_8 \).
First, the possible scenarios for Gracie's movement need to be established. If Gracie walks \( n \) blocks, and she is back home, she must take an equal number of steps east and steps west. Therefore, out of \( n \) steps, let \( k \) be the number of steps east, and thus \( n - k \) must be the same (steps west). This gives us the equation: \[ k = \frac{n}{2} \] which indicates that \( n \) must be even. For \( n = 8 \), we have \( k = 4 \).
To find the total number of valid sequences of 8 steps that lead back to the origin, we need to count how many ways we can arrange 4 heads (east) and 4 tails (west) in 8 flips of the coin. The number of such arrangements is given by the binomial coefficient: \[ \binom{8}{4} \]
Calculating this: \[ \binom{8}{4} = \frac{8!}{4!4!} = \frac{40320}{24 \times 24} = 70 \]
Next, we also need to consider the total number of possible paths she could take in 8 steps, where each step can either be east or west. The total number of sequences of 8 steps is: \[ 2^8 = 256 \]
Thus, the probability that she returns home after exactly 8 steps is the number of successful paths divided by the total number of paths: \[ p_8 = \frac{\binom{8}{4}}{2^8} = \frac{70}{256} \]
Now we can simplify \( \frac{70}{256} \): The greatest common divisor of 70 and 256 is 2: \[ \frac{70 \div 2}{256 \div 2} = \frac{35}{128} \]
Therefore, the probability that Gracie stops her walk after exactly 8 blocks is: \[ \boxed{\frac{35}{128}} \]