Suppose you are playing a game with a friend in which you bet ?dollars on the flip of a fair coin: if the coin lands tails youlose your ? dollar bet, but if it lands heads, you get 2? dollarsback (i.e., you get your ? dollars back plus you win ?dollars).
Let ? = \"the amount you gain or lose.\"
(a) What is the expected return ?(?) on this game? (Give youranswer in terms of ?)
Now, after losing a bunch of times, suppose you decide toimprove your chances with the following strategy: you will start bybetting $1, and if you lose, you will double your bet the nexttime, and you will keep playing until you win (the coin has to landheads sometime!).
Let ? = \"the amount you gain or lose with this strategy\".
(b) What is the expected return ?(?) with this strategy? (Hint:think about what happens for each of the cases of ?=1,2,3…flips).
(c) Hm ... do you see any problem with this strategy? How muchmoney would you have to start with to guarantee that you alwayswin?
(d) Suppose when you apply this strategy, you start with $20 andyou quit the game when you run out of money. Now what is ?(?)?
