A confidence interval is a net you throw at an unknown truth. Each fresh sample gives a new center and a new width — x̄ ± t·s/√n — so every pull lays a different trap across the same hidden seam.
The confidence level is a promise about the procedure, not the interval in your hand: run the trapline forever and about 95% of the traps will have caught the seam.
So a single 95% interval has no “95% chance” of holding the truth — it either did or it didn't. Reveal the seam and you'll see some traps sitting entirely to one side, catching nothing.
Raise the confidence and every trap widens; add samples and each trap tightens. The catch rate stays pinned to your promise either way — width buys precision, never certainty.
The trapline tangled mid-pull. The lesson continues without it — everything you have learned so far is safe.