jmc

Since Bin $A$ has one white ball and four black balls, the money ball has a $\frac{1}{5}$ chance of coming from Bin $W$ and a $\frac{4}{5}$ chance of coming from Bin $B$. The total expected value therefore is $E=\frac{1}{5}{E}_W+\frac{4}{5}{E}_B$, where $E_W$ and $E_B$ are the expected values of a ball drawn from bins $W$ and $B$, respectively. Since Bin $W$ has five 8 dollar balls and one 500 dollar ball, its expected value is $$E_W=\frac{5}{6}\times \$8+\frac{1}{6}\times \$500=\$90.$$Since Bin $B$ has three 1 dollar balls and one 7 dollar ball, its expected value is $$E_B=\frac{3}{4}\times \$1+\frac{1}{4}\times \$7=\$2.5.$$Therefore $$E=\frac{1}{5}{E}_W+\frac{4}{5}{E}_B=\frac{1}{5}(\$90)+\frac{4}{5}(\$2.5)=\boxed{\$20}.$$