Confidence Interval (CI) simulation in R: How?

I was wondering how I could check via simulation in R that the 95% Confidence Interval obtained from a binomial test with 5 successes in 15 trials when TRUE p=.5 has a 95% "Coverage Probability" in the long-run?

Here is the 95% CI for such a test using R (how can show that the following CI has a 95% coverage in the long-run if TRUE p = .5):

as.numeric(binom.test(x = 5, n = 15, p = .5)[[4]])