What does this equation mean?

Now suppose each individual tool call has some small, independent probability p of being consequential and wrong in a way a reviewer would have caught. The probability that at least one such call executes, unreviewed, inside a single gap of g calls is $\text{Exposure per gap} = 1-(1-p)^g$. This is the same “at least one” shape that shows up whenever independent trials are combined, applied here to the opposite question from the one it usually answers: not how many attempts are needed before one succeeds, but how many unwatched actions occur before one goes wrong. For small g , exposure grows almost linearly — each additional ungated action adds roughly p more exposure — so coarsening from asking every time to asking every fifth action multiplies exposure roughly fivefold while cutting interaction cost by the same factor: a real trade, not a free improvement in either direction. But the curve saturates: coarsening further, from a handful of calls to an entire multi-hour unattended task where g runs into the hundreds, buys comparatively little additional reduction in interaction cost per call, because the checkpoint count is already near its floor of one, while exposure is already close to its ceiling. Read against that shape, Claude Code’s default of asking per action, a plan-then-execute system’s single wide approval, and a background agent’s single end-of-task pull request are not simply “more” or “less” safe than one another in a straight line — they sit at different, deliberately chosen points on a curve that flattens.