Powerball is a multistate lottery. The following probability distribution represents the cash prizes of Powerball with their corresponding probabilities.

x (cash prize, $) ...........P(x)

Grand prize ...........0.00000000684

200,000 ........... 0.00000028

10,000 ............. 0.000001711

100 ............. 0.000153996

7 ............. 0.004778961

4 ............. 0.007881463

3 ............. 0.01450116

0 ............. 0.9726824222

(a) If the grand prize is $15,000,000, find and interpret the expected cash prize. If a ticket costs $1, what is your expected profit from one ticket?

(b) To the nearest million, how much should the grand prize be so that you can expect a profit? Assume nobody else wins so that you do not have to share the grand prize.

(c) Does the size of the grand prize affect your chance of winning? Explain.