During the lockdown, Viejas Casino has simplified their slot machines. They now have three reels, each with three symbols drawn on it: the number 7, a pair of cherries, and a bell. When you pull a lever, all three reels spin, and when they stop, a single symbol is displayed from each reel. If all three reels show the number 7, you win 6 dollars. If exactly two reels show a pair of cherries, you win 3 dollars. Otherwise, you win nothing.
Playing the machine (once) still costs 1 dollar.
(a) What is the probability you play the game and get three 7's?
(b) What is the expected profit (i. e., winnings minus cost) from playing a single game?
(c) With the knowledge from part (b), you go to management and suggest that they alter the slot machine to pay z dollars whenever three bells appear, so that the expected profit from a single game is zero. What amount x did you suggest?
(d) The casino managers do not listen to you. Instead, they change the slot machine so that now you win 1 dollar for getting three 7's, and otherwise you win nothing. It's a bit sneaky as at best you will break even (+1-1=0), but they count on us having worse things to worry about. After all of your gambling, you have just one dollar remaining. You sit down at the new slot machine and play until you run out of money. What is the expected number of games you play?