Given, Probability Mass Function:
$ => Pr[X_1 = 0] = Pr[X_1 = 1] = 1/2$ $… (1)$
$ => Pr[X_2 = 0] = Pr[X_2 = 1] = 1/2$ $… (2)$
$ => Pr[X_3 = 0] = Pr[X_3 = 1] = 1/2$ $… (3)$
Also it is given that random variable $Y=X_1X_2\bigoplus X_3$
We need to find: $Pr[Y=0 | X_3=0]$ = ?
$=> Pr[Y=0 | X_3=0]$
$=> Pr[ (X_1X_2\bigoplus X_3) = 0 | X_3=0]$ $(\because Y=X_1X_2\bigoplus X_3)$
$=> Pr[ (X_1X_2\bigoplus 0) = 0| X_3=0]$ (Put $X_3=0$ as it is given that $X_3=0$ then we have to find $Y=0$)
$=> Pr[ (X_1X_2) = 0| X_3=0]$ $… (4)$
It is given in the question that $X_i$ for $i=1,2,3$ are independent and identically distributed random variables
$=> Pr[ (X_1X_2) = 0]$ can be written as:
$=> Pr( X_1 =0 , X_2 = 0) + Pr( X_1 =1 , X_2 = 0) + Pr( X_1 =0 , X_2 = 1)$
Because we can say that $X_1 X_2 = 0 $ if either $X_1 = 0 $ or $X_2 = 0 $ or both $X_1 = 0$ and $X_2 = 0$
$=> [Pr( X_1 =0) \times Pr(X_2 = 0)] + [Pr( X_1 =1) \times Pr(X_2 = 0)] + [Pr( X_1 =0) \times Pr(X_2 = 1)]$
$=> (½ \times ½) + (½ \times ½) + (½ \times ½)$
$ => ¼ + ¼ + ¼ $
$=> ¾ $ (answer)