Brandon Lang is a creative entrepreneur who has developed a novelty soap item called Jackpot to target consumers with a gambling habit. Inside each bar of Jackpot shower soap is a single rolled-up bill of U.S. currency. The currency (rolled up and sealed in shrinkwrap) is appropriately inserted into the soap mixture prior to the cutting and stamping procedure. The distribution of paper currency (per 1000 bars of soap) is given in the following table.

Bill Denomination Number of bills

1 520

5 260

10 120

20 70

50 29

100 1

a. What is the expected amount of money in a single bar of Jackpot soap?

b. What is the standard deviation of the money in a single bar of Jackpot soap?

c. How many bars of soap would a customer have to buy so that, on average, he or she has purchased three bars containing a $50 or $100 bill?

d. If a customer buys 8 bars of soap, what is the probability that at least one of these bars contains a bill of $20 or larger?