GATE CS 2015 Set 3 Q37 — Probability: Random Variables and Distributions

We are given that $X_1, X_2, X_3$ are independent and identically distributed (i.i.d.) random variables with:

We need to find the conditional probability $Pr[Y=0 \mid X_3 = 0]$, where $Y = X_1X_2 \oplus X_3$.

Given the condition $X_3 = 0$, we can substitute $X_3 = 0$ into the expression for $Y$:

Since $X_3$ is independent of $X_1$ and $X_2$, the conditional probability simplifies to:

The product $X_1X_2 = 0$ if and only if at least one of $X_1$ or $X_2$ is $0$. Thus:

Since $X_1$ and $X_2$ are independent:

Therefore: