Suppose the 95th percentile confidence interval is $(a, b)$. Are the following two statements equivalent? If not, what is the difference?
Statement 1: There is a 95% chance that the interval $(a, b)$ contains the true (unknown) parameter.
Statement 2: There is a 95% chance that the true (unknown) parameter is between $a$ and $b$.
I saw a biostatistics lecture note that claims that Statement 1 is correct, while Statement 2 is incorrect.
EDIT: Additionally, the lecture note says that the uncertainty is associated with the confidence interval, not the true parameter. (This is a frequentist lecture taught by a Bayesian.)
EDIT2: Another example given in the context:
Incorrect statement: There is a 95% chance that Mozart was born between 1709 and 1799. Why? Mozart was born in 1756, and this fact does not change based on the estimation procedure.