Coverage probability

Coverage probability describes the method, not any single interval you have already computed. A 95% method has coverage probability 0.95, so if you drew 1000 random samples and built an interval from each, about 950 of those 1000 intervals would contain the parameter. Once you compute a specific interval such as 0.48 to 0.56, that interval either contains the parameter or it does not, so saying it has a 95% probability of containing the parameter is wrong. The correct statement is about the procedure before you sample: $P(\text{the interval covers the parameter}) = 0.95$.