Betting 20 dollars on a number across three dice

In a casino game you stake money on one number from $1$ to $6$, and then three fair dice are thrown. If your number shows on exactly one, two or three of the dice, you get your stake back plus one, two or three times the stake respectively; if it shows on none of the dice, the stake is lost. With a stake of $20, what is your expected profit from one play?

$-\tfrac{85}{54}$ dollars, about $-\text{\char36}1.57$. The number of matches is $\text{Bin}(3,\tfrac16)$, with probabilities $\tfrac{125}{216},\tfrac{75}{216},\tfrac{15}{216},\tfrac1{216}$ for $0,1,2,3$ matches. The expected profit per unit stake is $\tfrac{-125+75+30+3}{216}=-\tfrac{17}{216}$, and multiplying by $20$ gives $-\tfrac{85}{54}$.