This is actually essential to the problem. If you would only bet me if the coins came up heads, you could make all the coins heavily biased toward tales. In the rare scenario that 100 coins happened to come up anyways, you could show this to me to try to trick me into accepting, when you know that the next coin, like all the others, is guaranteed to be biased toward tales.
For actually solving the problem, I am no expert, but I think Laplace’s law of succession applies here. Laplace’s law states that when the only information that you have is the fact that there are only two possible results to a process (such as heads and tales) and the results to a number of trials that have already been done, the probability that the next result turns out a specific way is (s+1)/(n+2), where s is the number of times that is happened that way in the past and n is the total number of trials so far. I am not sure if that applies here because we may be working with a bit more information than this, but it might be correct.
This is actually essential to the problem. If you would only bet me if the coins came up heads, you could make all the coins heavily biased toward tales. In the rare scenario that 100 coins happened to come up anyways, you could show this to me to try to trick me into accepting, when you know that the next coin, like all the others, is guaranteed to be biased toward tales.
For actually solving the problem, I am no expert, but I think Laplace’s law of succession applies here. Laplace’s law states that when the only information that you have is the fact that there are only two possible results to a process (such as heads and tales) and the results to a number of trials that have already been done, the probability that the next result turns out a specific way is (s+1)/(n+2), where s is the number of times that is happened that way in the past and n is the total number of trials so far. I am not sure if that applies here because we may be working with a bit more information than this, but it might be correct.
In this case:
P(heads) = 101⁄102
EV(heads) = 101/102*V(heads) + 1/102*V(tails) = (101*$1000+1*$10)/102 = $101010/102 = $990 5⁄17
You can read more about Laplace’s law, including a derivation, in chapter 6 of Probability Theory: the Logic of Science by Edwin T. Janes.