How is the chance of at least one drop calculated?

Each attempt is independent, so the chance of getting nothing in N attempts is (1 minus rate) to the power N. The chance of at least one drop is one minus that. With a 20 percent rate over 10 attempts the odds are 1 minus 0.8 to the 10th, about 89 percent.

Why is it not just rate times attempts?

Multiplying rate by attempts double-counts overlapping successes and can exceed 100 percent, which is impossible. Binomial probability correctly accounts for the chance of getting two or more drops, so the true at-least-one odds are always below rate times attempts.

What is the expected number of drops?

Expected drops equals rate times attempts. This is the average you would get over many farming sessions, and it can be a fraction. A 20 percent rate over 10 runs averages 2 drops even though any single session might give 0, 1, 3, or more.

How many attempts reach a 90 percent chance?

Solve N = ceil(ln(1 minus target) divided by ln(1 minus rate)). For a 20 percent drop and a 90 percent target you need ceil(ln(0.1) / ln(0.8)) = 11 attempts. The tool reports the 50, 90, and 99 percent thresholds automatically.

Does this work for any drop rate?

Yes, for any independent per-attempt rate between 0 and 100 percent. Some Genshin materials have guaranteed or pity-style mechanics that this independent model does not cover, but most world-boss and overworld drops are well described by it.

Genshin Impact Drop Rate & Probability Calculator

Email me this result

Get this tool's output sent to your inbox, plus one useful tool a week. No spam, unsubscribe any time.

Multiplying a drop rate by your number of runs is the classic farming mistake — it overstates your odds and can exceed 100 percent. This calculator uses correct binomial probability to tell you the real chance of at least one drop, the average number you should expect, and how many attempts hit common confidence levels.

How it works

Attempts are independent, so the math is binomial:

P(at least one) = 1 − (1 − rate)^attempts
expected drops  = rate × attempts
attempts for p  = ceil( ln(1 − p) / ln(1 − rate) )

The last line inverts the at-least-one formula to tell you how many runs reach a 50, 90, or 99 percent chance.

Example and tips

A 12 percent drop over 15 runs gives a 1 − 0.88^15 ≈ 85 percent chance of at least one, with an expected 1.8 drops. To be 90 percent sure you would need ceil(ln(0.1) / ln(0.88)) = 18 runs. Remember the expected value is the long-run average — a bad streak of zero drops over many runs is unlucky but entirely possible.