How to apply confidence intervals to probability?

Hello. So the question is there are 200 people who are all making 95% confidence intervals (using the same data and etc) and determining if X falls within that confidence interval.





1) How many people are expected to find that X falls within their interval.





Is it 200*0.95 = 190?





2) What is the probability that all 200 students get X in their interval?





I was thinking that it is 0.95^200 but that can't be right.





Any help would be greatly appreciated.How to apply confidence intervals to probability?
1) If you are to run a large number of confidence intervals, each based on a separate estimate for the mean (with confidence level (1 - alpha)), then yes, the expected number of confidence intervals containing the true mean would be n * (1 - alpha), or 200 * 0.95 like you said.





In general, this means that any future confidence interval (the sample has not yet been measured or tabulated) has a 95% chance of containing the true mean. It DOES NOT mean that an interval that has already been run has a 95% chance. Probability only applies to events which haven't happened yet.





2) How to find the probability that all 200 students will get the true mean in their interval is a binomial distribution question. Sample size is n, probability of success, p, is 0.95. So, the probability of 200 independent successes would be (0.95) ^ (200), or 0.00003505, or 0.003505%. Even the most likely outcome, exactly 190 successes and 10 failures, has the relatively small probability of .1284, or 12.84%.





With 201 possible outcomes, no one of them will have a very large possibility. 190-10 is just the EXPECTED outcome.