(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

There are 2 coins in a bin. When one of them is flipped it lands on heads with probability 0.6 and when the other is flipped it lands on heads with probability 0.3. One of these coins is to be randomly chosen and then flipped. Without knowing which coin is chosen, you can bet any amount up to 10 dollars and you then either win that amount if the coin comes up heads or lose if it comes up tails. Suppose, however, that an insider is willing to sell you, for an amount c, the information as to which coin was selected. What is your expected payoff if you buy this information? Note that if you buy it and then bet x, then you will end up either winning x - c or -x - c (that is, losing x + c in the latter case). Also, for what values of c does it pay to purchase the information?

2. The attempt at a solution

I'm asked to find E[Y] where Y is a discrete random variable representing the payoff given that I've paid c to buy information. What are the possible values Y can take? x can be at most 10 - c so -(10 - c) - c = -10 <= Y <= (10 - c) - c = 10 - 2c.

If Y >= -c, then the coin must have landed heads. Otherwise, it must have landed tails. Thus, P{Y >= -c} equals the probability of the coin landing heads, P(H), and P{Y < -c} equals the probability of the coin landing tails, P(T).

Let p(y) = P{Y = y}. There are 11 - c values of y for which y >= -c and 10 - c values of y for which y < c. Thus p(y) = P(H) / (11 - c) for y >= -c and p(y) = P(T) / (10 - c) otherwise.

E[Y] = P(T) / (10 - c) * (-10 + ... + -c - 1) + P(H) / (11 - c) * (-c + ... + 10 - 2c)

Is this correct?

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# Homework Help: Expected Payoff Given Info.

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