An event A has n possible outcomes, which have the values x1, x2, . . . , xn . The probabilities of the n outcomes occurring are p1, p2, . . . , pn. The expected value V of an event A is the sum of the products of the outcomes' probabilities and their values,
V = p1x1 + p2x2 + . . . + pnxn.
(a) To win California's Super Lotto Plus game, you must match five different numbers chosen from the numbers 1 to 47, plus one MEGA number chosen from the numbers 1 to 27. You purchase a ticket for $1. If the jackpot for the next drawing is $12,000,000, what is the expected value of the ticket?
(b) You are playing a dice game in which you need to score 60 points to win. On each turn, you toss two six-sided dice. Your score for the turn is 0 when the dice do not show the same number. Your score for the turn is the product of the numbers on the dice when they do show the same number. What is the expected value of each turn? How many turns will it take on average to score 60 points?