How to Use the Upgrade Stage Simulator
Knowing your success rate alone doesn't tell you how many attempts you'll actually need on average to hit your goal. This simulator takes your success rate, the number of stages needed, and the cost per attempt, then shows not just the average (expected) number of attempts, but also how many tries and how much it would cost in a lucky run versus an unlucky run.
Under the hood, the number of attempts needed to clear a single stage follows a geometric distribution, and clearing several stages in a row follows a negative binomial distribution. Instead of actually running tens of thousands of simulated upgrades, this calculator uses the mean and standard deviation formulas of that distribution to approximate the top 10% (lucky) and bottom 10% (unlucky) ranges instantly and accurately.
A lower success rate produces a larger standard deviation, which widens the gap between the lucky and unlucky cases even further. When planning for a low-probability upgrade, don't rely on the average alone — budget with the unlucky case in mind for a safer plan.
Frequently Asked Questions
This calculator uses the statistical fact that the number of attempts follows a negative binomial distribution, so it computes the mean and the top/bottom 10% range instantly with exact formulas instead of running tens of thousands of trials.
Using the standard deviation around the mean, it approximates the top 10% (lucky) and bottom 10% (unlucky) attempt ranges. The lucky case can never be fewer attempts than the number of stages needed.
Yes — a lower success rate increases the standard deviation, so the gap in attempts and cost between the lucky and unlucky cases becomes much larger.