Consider a casino game in which a wheel with 11 spots numbered 0-10 is used. At every turn any of the 11 numbers can come up with equal probabilities. There are three bets available with the following payoffs for a bet of $1 (where x is the outcome of the wheel).,Bet 1: If x turns out to be 0, you lose $9. Otherwise you win $1.,Bet 2: If x turns out to be 1-5, you lose $1. Otherwise you win $1.,Bet 3: If x turns out to be 6-10, you lose $1. Otherwise you win $1.,(a) What are the expected values and variances of the three bets?,For the next questions assume that you can bet $1000 once.,(b) How would you allocate the $1000 if you can bet only on Bet 1 and Bet 2 with the goal of having minimum variance?,(c) Can you make a complete arbitrage (that is, guarantee a positive profit regardless of the outcome) by some allocation of the $1000 among the three bets? Explain your answer.
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