Confidence Interval has no relation to the probability? An Intro to Stats class has the following problem:
Find and interpret the 90% confidence interval for the true mean

The provided answer is this:
The probability is either 1 or 0. 
Its either true or it isn't! No maybes.

This answer confuses me. I thought the 90% confidence interval means there's a 90% probability that the mean is in that interval. Does the confidence interval really have no bearing on the probability?
Can someone explain why the provided answer makes sense at an introductory level?
I've looked at this question, but it doesn't seem to answer my question at all.
 A: In conventional frequentist statistical inference, a quantity is considered "random", and therefore has a probability distribution assigned to it, only if it changes when a new sample is taken.  "Sample" does not mean a single observation; it means a set of observations on which inferences are based.  You randomly take 50 men out of a population.  That's a sample.  Toss the back and grab another 50; that's another sample.  Each time you take another sample of 50, you get a sample mean and a sample standard deviation.  $90\%$ of the time when you do that, you compute a confidence interval that contains the population mean.  So the probability that your next confidence interval contains the population mean is $90\%$.  But once you've got those two numbers --- the upper and lower bounds of the interval, can you say that there's a $90\%$ chance that the population mean is between them?    Say the numbers are $68$ and $71$, and this is a $90\%$ confidence interval for the heights in inches of 21-year-old men based on the sample you took of $50$ of them.  Will the population mean be between $68$ and $71$ in $90\%$ of all samples of $50$?  No, it won't.  The population mean will be between $68$ and $71$ every time you take a sample of 50, or else it will fail to be between $68$ and $71$ every time you take a sample of 50.
In Bayesian, as opposed to frequentist, inference, one may assign a probability distribution to any quantity that is uncertain, regardless of its not changing from one sample to the next, and thus one may find a conditional probability distribution of the population mean given the observations in a sample.  But such intervals are not normally called confidence intervals; they are called posterior probability intervals or credible intervals.
The Bayesian-versus-frequentist issue is not a disagreement about mathematics, but rather about the way in which mathematics should be applied.
A: The probability at stakes makes the following assumption:
the sample $X_1,\dots, X_n$ follows a law such as $X_i\sim X$ and are independent.
The probability given in a confidence interval is such as the true mean is in the interval with a probability given the sample AND the previous assumption of 90%.
In other words, if you try with a different sample over and over again, the frequence at which the true mean is in the confidence interval converges to 90%.
A: Michael Hardy gave a nice answer in terms of fequentist vs. Bayesian interpretations of what events can be assigned probabilities [+1 :)]. I would like to offer a complementary explanation which may help you as well.
Like you, I struggled with the interpretation of confidence intervals when I first learned them. Supposedly, we are supposed to have a relatively high degree of trust that a 99% confidence interval contains the true value of the parameter. However, then the professor turns around and basically undermines this intuition by saying "NO...it's either right or wrong!" Now, it sounds like we've gained nothing from forming this so-called "confidence interval". My first reaction was to lose confidence in confidence intervals!
I was able to revive my trust in these intervals by realizing that this whole thing about a particular interval being right or wrong, while correct in an objective sense, does not really help in real-world inference. We don't know which (otherwise, why do the experiment), so we must ascribe a number to the interval that tells us something about its reliability. This is the "confidence", which is a type of probability. For example, a 99% CI contains the true value 99% of the time (over repeated applications of the procedure for creating CIs...not this particular CI). The probability is a statement about the overall methodology, not a particular interval.
Hence, if you were the betting type, you would always bet that a particular 99% CI contains the true parameter value, since in 99% of the possible realizations of this experiment, the resulting CI will contain the true value, and you don't know which realization you've actually seen.
As you can see, the confidence is a well defined probability, but it is used to evaluate a procedure, not an individual interval. We trust the actual interval because it was generated by a procedure that is correct 99% of the time, on average.
An alternative way to think about it is to view a 99% CI as an "almost perfect" oracle. Out of every 100 questions you ask the oracle, 99 of them will be correct. When you calculate a 99% CI, you are effectively asking the oracle: "Give me an interval that contains the true mean", and the actual interval is the response. Since you know the oracle has a "track record" of being 99% correct, you have a very high level of confidence that you can trust this interval to contain the mean.
The key distinction between the confidence assigned to an interval and a "normal" probability assigned to a random variable is in the nature of the probability -- is the probability Objective or Subjective?


*

*A probability is "objective" [note my use of quotes...its debatable how truly objective this type of probability is (see reference sequence problem)]if it represents the long-run frequency of the occurrence of the event. (Frequentist definition of probability) Think an infinite sequence of coin tosses for a fair coin...the "Objective probability" is 1/2.

*A probability is "subjective" if it represents a degree of trust (note how abstract this is). It is essentially an abstract representation of a feeling. For example, "What is the probability that there is life on Europa?" There is no sample space that applies to this question, as there is only 1 Europa, so either the question is true or false...but we don't necessarily feel that both options are equally likely. The subjective probability reflects this.
Therefore, you can think of confidence as an objective probability only if you see it as embedded in an infinite series of similar experiments, where each time you re-calculate the confidence interval. In this case, you are saying "If conclude that the true value is within this particular interval, then if I repeated this decision for a large number of repeated experiments of this type, I will be right most of the time. I like those odds, so I will conclude that this interval contains the true value."
However, we could easily say this subjectively as "Of all the possible 99% CIs that could 
be formed from this type of experiment, 99% will be correct, hence I am 99% sure that this interval contains the true value (basically acknowledging your lack of complete certainty)"
So, as you can see, your book is correct...but the tendency to belabor this point in intro stats is not helpful, as it undermines the intuition about why we care about CIs in the first place. Just think of confidence as a subjective probability and you will be fine :-)
Note/Aside
There is an oft-used example about what constitutes a probability that I would like to address, as you may come across it in the future:
Your friend tosses a fair coin and hides the result...what is the probability that it is heads?


*

*Frequentist/objective probability answer: This event does not have a probability, it is either heads or tails.

*Bayesian/Subjective probability answer: 50%


Now, lets change the timing:
Your friend will toss a fair coin...what is the probability that it is heads?


*

*"Alleged" Frequentist/objective probability answer: 50%

*Bayesian/Subjective probability answer: 50%


I say "Alleged" because I don't think it is totally correct: The next toss will either be heads OR tails, so how are we assigning a probability to this occurrence?  Referring back to the definition of "objective" probability, a probability applies to an infinite sequence of tosses, hence, if the event is "coin shows heads", then we can correctly embed that event in an infinite sequence of tosses and conclude that the probability "coin shows heads" is 50%. But we are not asking about the generic probability of a head, we are asking about the probability that this toss will result in heads...there is only one chance for this particular toss to occur, so I would argue (somewhat pedantically, perhaps ;-P) that this singular event does not have a frequentist probability. If it does, it seems to give undue influence to the timing of an event that does not convey any information about the actual result. 
If the above didn't make sense, then just ignore it. 
