Suppose $X_i$ for $i = 1,2,3$ are independent and identically distributed random variables whose probability mass functions are $\Pr[X_i = 0] = \Pr[X_i = 1] = 1/2$ for $i = 1,2,3$. Define another random variable $Y = X_1 X_2 \oplus X_3$, where $\oplus$ denotes XOR. Then $\Pr[Y = 0 | X_3 = 0] =$ _________.

We need to calculate the conditional probability $\Pr[Y = 0 \mid X_3 = 0]$.

Given the definition of $Y$:

Conditioned on $X_3 = 0$, we have:

Thus:

Since $X_1, X_2,$ and $X_3$ are independent random variables, the distribution of $X_1 X_2$ is independent of the condition $X_3 = 0$:

Using the complement rule:

Since $X_1 X_2 = 1$ occurs if and only if $X_1 = 1$ and $X_2 = 1$:

Therefore: