1.You go to a casino with $31 and the following strategy: In thefirst round you bet $1, if you win, you receive double your bet,you leave. If you lose you double your bet to $2. You continue thisstrategy that you receive double your bet and leave if you win, anddoubling you bet in the previous round if you lose, until you runout of money. Let X equal the total amount of money you win(”winning” a negative amount of money is the same as losing thatamount). If the probability of winning in each round is independentand equal to 9/20:
(a) What is the pmf, pX(x), of X?
(b) What is E[X] and Var[X]?
2. You are dealt one card from a full deck of 52 cards and youropponent is dealt two cards (without replacement). If you get aface card (jack, queen or king) your opponent pays you 5, if youget an ace your opponent pays you 2. If you don’t have a face cardor an ace, but you have a spade and your opponent doesn’t have aspade they pay you 1. In all other cases you pay 1. What is theexpectation of your winnings?
