3. Suppose we play the following game. I give you $100 for your initial bankroll. At each time n, you decide how much of your current wealth to bet. You cannot borrow money. You can only play with the money I gave you in the beginning or any money that you have won so far. The game is simple. At each time n 2 1, you decide the amount to bet. I will roll a fair die. If the die comes up 1,2,3, . . . , or 5, you win; if the die comes up 6, then you lose. IOW, if you bet $10 on the rst roll, you will either have $90 or $110 after the rst roll. ' (a) Suppose you wish to maximize your prot on the rst roll. How much should you bet? (Most of you will get this wrong.) (b) What is the expected prot on the rst roll if your bet is b with O S b S 100? (0) Suppose you wish to maximize your expected prot on the rst roll. How much should you bet? (d) Suppose you wish to maximize your expected prot betting on the nth roll. How much of your current wealth do you bet? (6) Let X\" be your wealth after H plays; thus, X0 = $100. What is your ex pected wealth after 12 plays if you always bet to maximize your expected , prot? (f) Assuming you maximize your expected prot each time, your expected wealth converges to what as n > 00. (Sounds good, no?) , (g) What happens to your wealth as n ) co