A box contains ten $1 bills, five $2 bills, three $5 bills, three $10 bills, and three $100 bills. A person is charged $20 to reach into the box while blindfolded and pull out one bill. Find the person's expected average winning or profit each time the game is played. For this game to be fair, how much should the person be charged to play?

(Hint: Be sure to work this problem using "Expectation". There are a couple of ways that you can "play" with the numbers to get the following correct answer. But, the "method of solution" is very important, and you should use "Expectation". Here are the answers... but how do you get them? The player should expect to lose $4.79 per play (on average). The game should charge $15.21 to be fair. A "fair game" is when everyone "breaks" even in the end; neither party gains money!)