ZZZ Pity Calculator

Three inputs drive the whole result. Channel changes the formula entirely since each has its own cap and rates: the Agent channel (standard + limited), the W-Engine channel, and the Star-Studded Cast (standard/stable) channel. Current pity is how many pulls in a row you've made without an S-rank — it counts from your last S-rank and resets to 0 whenever you hit one. The guarantee switch only applies to the limited channel: flip it on if your last S-rank was NOT the featured unit (you lost the 50/50), because your next S-rank there is then guaranteed to be the one you want.

Read the output in order: the chance of the exact target within budget, the chance of any S-rank (including off-banner), the average pulls needed to hit it, and the worst-case safety net. The distribution chart shows cumulative probability climbing pull by pull — it steepens sharply around the soft pity zone and goes nearly vertical at the hard cap. The odds-by-milestone table lets you compare a few budget scenarios at a glance without re-entering numbers.

Each channel has its own separate hard cap: the Agent channel (both standard and limited) guarantees an S-rank within 90 pulls; the W-Engine channel guarantees one within 80 pulls. These are the in-game published figures, not estimates — hit the cap and it's 100% guaranteed, even though the base rate per pull sits around 0.6% (Agents) or around 1% (W-Engines, the highest weapon base rate among HoYo games).

The Star-Studded Cast channel — home to permanent Agents and W-Engines — has its own separate milestone: reach 300 pulls there and you freely SELECT any standard S-rank Agent, no luck involved. It also guarantees at least one A-rank or better every 10 pulls in a row, so this channel rarely leaves you empty-handed for a long stretch the way the other two can.

When you land an S-rank on the limited Agent channel, there's a 50% chance it's the featured unit and a 50% chance it's a random standard Agent. Lose that coin flip and the guarantee kicks in: your next S-rank on that channel is then 100% guaranteed to be the featured one. So the worst case to secure a limited Agent is 90 pulls (loss) plus 90 more (guaranteed win) — 180 pulls, or 28,800 Polychrome.

The W-Engine channel is much friendlier: the featured W-Engine lands at 75% right on the first S-rank (75/25 instead of 50/50), so the worst case is still 80 plus 80 for 160 pulls, but the odds of ever needing that full stretch are far lower since the first-try win rate is already high. The tool automatically switches to the right guarantee math when you change channels, so you never have to remember which is 50/50 and which is 75/25.

If the S-rank rate stayed at a flat 0.6% from pull 1 to pull 90, most players would rarely feel close to the hard cap since cumulative odds would already be decent early on. In practice the game uses a rising curve called "soft pity": starting around pull 74-75, the per-pull S-rank rate jumps in steps instead of staying at the base rate, climbing faster the closer you get to 90 until it hits a guaranteed 100%. That's why most players actually land their S-rank somewhere around pull 75-85, rarely needing to reach the literal 90th pull.

The exact soft pity onset isn't officially published — community measurements put it somewhere around pull 74 to 75 depending on sample size, with the rate climbing roughly 2% per pull past that point. This is a large-sample community statistic, not a figure from official documentation, so the tool flags it as a community estimate rather than an absolute claim — transparency over false precision.

To know how much to save before a new banner drops, work backward from the worst case: multiply the pulls needed (up to 180 for a limited Agent starting at 0 pity with no guarantee) by 160 Polychrome per pull. But most players don't need to bank the full worst case — plug in your actual pity and guarantee status, check the probability at a few budget milestones (say 30, 60, 90 pulls), and decide how much risk you're comfortable carrying.

Three practical saving tips: if you already hold the guarantee (you lost the 50/50 last time), the upcoming limited banner is nearly a sure thing at the cap — prioritize saving for that one first. If the tool already shows 80-90%+ at your current budget, saving further adds very little for a lot more farming. And don't forget: switching between "how many pulls" and "how much Polychrome" is just a display toggle in the tool — both modes describe the same odds, just a different unit to track your wallet.

Hand-counting with "on average 1 in 0.6% pulls gets an S" only holds if the rate never changes — but since soft pity spikes the rate after pull 74-75, the real average pull count is well below the naive 1/0.6% ≈ 167 figure. That's why community-measured averages vary quite a bit — some report roughly 62 pulls counting from 0 pity including a possible loss-then-guaranteed-win, others report 75-80 pulls if only averaging near-cap outcomes — the gap comes from how each source defines "average," not a calculation error.

The tool uses a closed-form probability model: it computes the entire soft-pity curve plus guarantee state exactly with math, rather than running tens of thousands of trials and averaging like some Monte-Carlo simulators do. That approach gives stable results — feed it the same numbers and it returns the exact same probability every time, with no simulation noise.