You're thinking of trying a variant of Chuck-a-luck. In this variant, you choose one number in the range $1$ to $6$ and five dice are rolled. These are the payouts:
Number of dice equal to your Your number payout --------------------- None - 5 USD 1 0 USD 2 + 5 USD 3 +10 USD 4 +15 USD 5 +20 USD
So for example if none of the dice show your number, you pay $\$5$, while if exactly 1 die shows your number, you don't pay or receive anything.
You suspect the odds are against you but you want to know exactly how much. You've also had a few too many, so you don't want to have to resort to inclusion-exclusion or anything complicated.
What is a quick and simple way to calculate the expected value of this game?