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# Probabilities - investment science

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9. There are two propositions: (a) I flip a coin, If it is heads, you are paid \$3; if it is tails, you are paid \$0. It costs you \$1 to participate in this proposition. You may do so at any level, or repeatedly, and the payoffs scale accordingly. (b) You may keep your money in your pocket (earning no interest). Here is a third proposition: (c) I flip the coin three times. If at least two of the flips are heads, you are paid \$27; otherwise zero. How much is this proposition worth?

17. There is a market for bets on the outcome of a coin toss. The possible outcomes are heads, tails, and edge. There are three assets traded in that market:

Asset A pays \$1 independent of the outcome
Asset B pays \$1.50 for a head and \$0 for tails and \$ if the coin lands on its edge
Asset C pays \$10 if and only if the coin lands on its edge (and \$0 otherwise)

The prices of those assets are always constant and are fixed at \$1. We assume that the payments to the winning bidders take place immediately after the outcome is determined.
(a) What is the implied risk-free interest rate in this market?
(b) What are the risk-neutral probabilities of each of the possible coin toss outcomes (heads, tails, edge)?