Help - translating confidence interval into probability

My company makes plastic vascular conduits used to replace diseased blood vessels in the body. I perform tests on product, stuff like stretching it until it breaks and pressurizing it until the device sweats. I usually take 3 samples from one lot then report the confidence interval on my results. I understand that for (example) a 95% CI, I am 95% confident the population (i.e. total lot) mean falls within my confidence interval.

I understand that a confidence interval calculated CI = mean +/- z x stdev/n.5 and is the integral of a Bell/gauss curve +/- z*stdev (z is z-score, n is sample size).
I am trying to figure out how to report the probability a sample from my population will fall outside my CI. It is NOT 5% for 95% CI, I know that is a common misconception. Can someone help me understand?


Not a robit
Frequentist based confidence intervals IMHO are more confusing than the mystical p-value. The general description I use in my head is that upon repeated sampling, 95% of constructed confidence intervals will include the true value.