Note: apologies in advance if this is a duplicate, I didn't find a similar q in my search
Say we have a true parameter p. A confidence interval C(X) is a RV that contains p, say 95% of the time. Now suppose we observe X and compute C(X). The common answer seems to be that it is incorrect to interpret this as having a "95% chance of containing p" since it "either does or it doesn't contain p"
However, let's say I pick a card from the top of a shuffled deck and leave it face down. Intuitively I think of the probability of this card of being the Ace of Spades as 1/52, even though in reality "it either is or it isn't the Ace of Spades." Why can't I apply this reasoning to the example of the confidence interval?
Or if it is not meaningful to talk of the "probability" of the card being the ace of spades since it "is or it isn't", I would still lay 51:1 odds that it isn't the ace of spades. Is there another word to describe this information? How is this concept different than "probability"?
edit: Maybe to be more clear, from a bayesian interpretation of probability, if I'm told that a random variable contains p 95% of the time, given the realization of that random variable (and no other information to condition on) is it correct to say the random variable has a 95% probability of containing p?
edit: also, from a frequentist interpretation of probability, let's say the frequentist agrees not to say anything like "there is a 95% probability that the confidence interval contains p". Is it still logical for a frequentist to have a "confidence" that the confidence interval contains p?
Let alpha be the significance level and let t = 100-alpha. K(t) be the frequentist's "confidence" that the confidence interval contains p. It makes sense that K(t) should be increasing in t. When t = 100%, the frequentist should have certainty (by definition) that the confidence interval contains p, so we can normalize K(1) = 1. Similarly, K(0) = 0. Presumably K(0.95) is somewhere between 0 and 1 and K(0.999999) is greater. In what way would the frequentist consider K different from P (the probability distribution)?