Edit: Revising original question
I'm trying to make sure I understand frequentist confidence intervals. Below is a hypothetical setup that illustrates the way I'm trying to think about this, and I'm wondering where/if I'm going wrong?
Setup: A computer generates 100 numbers from a binomial distribution, B(100,p) with the parameter p only known to the computer. The computer also calculates a clopper-pearson 95% confidence interval along with the data. There is an exchange on which the following contract can be bought and sold: "This contract pays out \$1 if the confidence interval contains p." At the end of the week, the computer reveals p to all market participants so that the contract can resolve.
My question relates to the fair value of the contract. BEFORE the numbers are generated, clearly the contract has a fair value of \$0.95. AFTER the numbers and interval are generated, unless there was something obviously absurd about the confidence interval, wouldn't the fair value still be roughly be \$0.95? So that in the presence of uncertainty, it's often reasonable to think about a 95% confidence interval as having a 95% chance of containing the parameter? And I've seen Jaynes in his book give examples where the confidence interval is obviously wrong, but most of the time that doesn't seem to be the case (if there was not much uncertainty, wouldn't there be no role for statistics?)
The "fair value" of the contract in this setup is how I am thinking about probability and is a proxy for the "probability of containing the parameter."