Expected Sum of Bills Drawn

Total value in the box: $4\cdot 10+3\cdot 20+100=\$200$ over $8$ bills, so the mean bill is $\$25$.

Label the bills $1,\dots,8$ with values $v_1,\dots,v_8$. Let $I_j$ indicate that bill $j$ is drawn. By symmetry each bill is drawn with probability $\mathbb{P}(I_j=1)=\frac{7}{8}$. The total drawn is $S=\sum_j v_j I_j$. By linearity,

Equivalently, each of the $7$ draws is marginally a uniform bill, with mean $\$25$, so $\mathbb{E}[S]=7\cdot 25$.

Alternative. Exactly one bill is left behind, and it is uniform on the eight. Its expected value is $\frac{200}{8}=25$, so $\mathbb{E}[S]=200-25=175$.

Key idea. Each bill is drawn with probability $\frac{7}{8}$, so linearity gives $\frac78$ of the total without any joint distribution.