Since Bin $ A $ has one white ball and four black balls, the money ball has a $ \dfrac{1}{5} $ chance of coming from Bin $ W $ and a $ \dfrac{4}{5} $ chance of coming from Bin $ B $. The total expected value therefore is $ E = \dfrac{1}{5}E_W + \dfrac{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 \[ E = \frac{1}{5}E_W + \frac{4}{5}E_B = \frac{1}{5}(\$ 90) + \frac{4}{5}(\ $2.5) = \boxed{\$20}.\]