no prime digits

Let the expectation be $E$ . Pick two random numbers $\displaystyle x=\sum_{i=1}^\infty\frac{x_i}{10^i}$ and $\displaystyle y=\sum_{i=1}^\infty\frac{y_i}{10^i}$ , with $x_i, y_i\in \{0, 1, 4, 6, 8, 9\}$ .

Case 1: they start with the same digit. The expectation of $|x-y|$ is then $\frac{1}{10}E$ .

Case 2: they start with different digits. Suppose WLOG $x>y$ . The expectation is $\displaystyle E(x-y|x>y)=\sum_{i=1}^\infty\frac{x_i-y_i}{10^i}=\frac{1}{10}E(x_1-y_1|x_1>y_1)+\sum_{i=2}^\infty\frac{1}{10^i}E(x_i-y_i)$

$\forall i, E(x_i-y_i)=E(x_i)-E(y_i)=0$ .

$E(x_1-y_1|x_1>y_1)=\frac{1+4+6+8+9+3+5+7+8+2+4+5+2+3+1}{15}=\frac{68}{15}$

Case 1 has probability $\frac{1}{6}$ , case 2 has probability $\frac{5}{6}$ .

$E=\frac{1}{6}·\frac{1}{10}E+\frac{5}{6}·\frac{1}{10}·\frac{68}{15}$ so $\frac{59}{60}E=\frac{68}{60·3}$ and therefore $E=\frac{68}{177}$ and $68+177=\boxed{245}$ ,