1/1 110% + $03 State your conclusions in typed complete English sentences and show your work coherently. For the optimal bet question you may use the general 1 formula and the simple Kelly formula if applicable. 1. The game is to ip a fair coin. If you get heads, you win a dollar, if tails, you lose a dollar. Then let X be your win. Check that E(X) = 0 and nd crl. Next assume that the coin is not fair, say p = P(heads) = .6 and q = P(tails) = .4. Find ,u and a. 2. Suppose you play a lottery with payout of 100 dollars for each dollar bet where you have a one percent chance of winning. Find p. and or for a one unit bet. ' 3. Suppose you play Mega Millions lottery with payout of 300 million dollars for each dollar bet Where you have a one in 176 million chance of winning. Find 1.: and a for a one unit bet. 4. Suppose you toss a tetrahedral die with 4 equally likely sides a,b,c,d and you win 2 units for a, lose one unit for b and tie (push) for ad. Find the the expected value and variance of a unit bet. a. 5-8. Suppose you have a $10, 000 bankroll. Find is the Kelly optimal bet in 1, 2,3,4, rounding to the nearest dollar? . 9. Find the standard deviation of a one dollar European Roulette bet. For European roulette, a single number wins 1/37 and loses 36/37. The expected value is 1$ + 3551? = 0.027 for each unit bet (2.70% house edge). What is your optimal bet for this game? What is the optimal bet from the house's point of view, assuming a house bank of 10 million dollars? 10. Find the optimal bet for the house in American Roulette as a percentage of bankroll, where the gambler bets on red. The bet pays even money 1:1. ' TIP: H: ' '4 . ,2\" . 1 _- . r-W GE .9 an iiiili