Background: The Daily 3 game is a daily game, drawn every day except for Saturday and Sunday. It consists of three sets of balls, each numbered from $0$ through $8$ ($9$ is omitted due to its visual similarity to $6$). One ball is drawn from each set giving a $3$-digit winning number. All numbers must be in the correct order to win. A ticket costs $\$1$. The prize for getting all three numbers correct is $\$500$.
Question: Assuming each possible outcome is equally likely to appear, how much money, on average, will the Agency receive from each Daily 3 ticket, after paying winners?
I think this is an expected value question. Here is my work so far: $X$ = Agency's profit. Then $$E(X)= P(\text{Grand Prize})\cdot(-500 + 1) + P(\text{Losing Ticket})\cdot(1).$$
$P(\text{Grand Prize}) = 1/729.\;$ $P(\text{Losing Ticket}) = 728/729.\;$ So $$E(X) = 1/729 \cdot (-500 + 1) + 728/729 \cdot 1 = .314. $$
Answer: Agency's Profit is $.31$ cents per ticket (this seems too low, I think I did something wrong)
