(Solution) - Now place yourself exactly in the same setting as before -(2025 Original AI-Free Solution)

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Now place yourself exactly in the same setting as before, where the market quotes the above R. It just happens that you have a close friend who offers you the following separate bet, R?:

The random event behind this bet is the same as in R. Now consider the following:
(a) Using the R and the R?, construct a portfolio of bets such that you get a guaranteed risk-free return (assuming that your friend or the market does not default).
(b) Is the value of the probability p important in selecting this portfolio? Do you care what the p is? Suppose you are given the R, but the payoff of R? when the incumbent wins is an unknown to be determined. Can the above portfolio help you determine this unknown value?
(c) What role would a statistician or econometrician play in making all these decisions? Why?