Gacha Pity Calculator

About this tool

A pity system changes the arithmetic completely, and almost every "how many pulls" answer on the internet gets it wrong by ignoring that. If a 5-star has a 0.6% base rate, dividing one by 0.006 gives 166.7 pulls. That is the right answer for a game with no pity at all. With a hard pity at 90 pulls, no single attempt can run past 90, so the true average has to be far lower — around 62 pulls with the widely used soft-pity curve.

The reason the average sits so far below the halfway point of the range is that the chance is not flat. It is 0.6% for the first 73 pulls, then climbs steeply: 6.6% at pull 74, 12.6% at 75, and so on until it reaches 100% at 90. Because the climb is concentrated in the last seventeen pulls, most successes land in that narrow window, and the few unlucky runs that reach the cap are too rare to drag the average far up.

The model here is the one the community has settled on for Genshin-style banners: 0.6% base, soft pity from pull 74, +6 percentage points per pull, hard pity at 90. The developer has never published those numbers — only the blended 1.6% figure — so treat them as a good working model rather than gospel. Every input is editable, which is the point: the same maths covers a 1% rate with a 100-pull cap, a 3% rate with no soft pity, or a banner whose rules you found in a patch note.

How the expected value is actually computed

The expected number of pulls is the sum of the survival probabilities: for each pull, the chance that you have still not got a 5-star before it. Add those up from the first pull to the cap and you have the average. It is not one divided by the base rate.

The survival form is easier to trust than the textbook formula. Before the first pull the chance of having nothing is 100%, so the first pull contributes 1 to the total. Before the second it is 99.4%, so it contributes 0.994. Each pull adds however likely it is that you are still empty when it starts. The sum stops at the hard pity, where the last pull contributes only the small probability of having missed everything before it.

This is the same number you would get by enumerating "first success on pull 1, on pull 2, on pull 3" and multiplying each by its pull number, but it needs no extra bookkeeping and no subtraction of nearly equal numbers. It also makes the effect of the cap obvious: every pull after the cap would have added to the total, and the cap deletes them all at once.

A sanity check that catches most mistakes: with no soft pity, a flat 1% rate and a 100-pull cap, the formula gives 63.4 pulls, not 100 and not 100/2. The closed form is (1 − 0.99^100) / 0.01, and the cap is what turns 100 into 63.4.

Why the median is not the average

Half of all players finish before the median, so it answers "when does a typical player get one" rather than "what does the long-run average look like". With the default model the median is pull 76 while the average is 62.3.

A skewed distribution has two different centres. The average is pulled down by the lucky runs — someone who hits on pull 5 counts just as much as someone who grinds to 89, but contributes far less to the total. The median ignores how early the early ones were and only counts how many finished. When a large mass of players finish in the first twenty pulls, the average falls well below the halfway point.

For the default model the 90th percentile is pull 80 and the 99th is 83. That shape is the signature of a soft-pity curve: almost nobody reaches the cap, because by pull 89 the chance is already 96.6%, so the probability of arriving at 90 empty-handed is around seven in a hundred million.

Both numbers are worth showing, because they answer different questions. "How much should I budget" is an average question. "Am I unlucky" is a percentile question — and a player at pull 76 is exactly at the middle, not behind.

What the soft-pity curve does to the last twenty pulls

Soft pity turns a flat lottery into a ramp. In the default model the chance is 0.6% up to pull 73, then 6.6% at 74, 12.6% at 75, adding six points each time until 100% at 90. More than half of all 5-stars arrive inside that final stretch.

The ramp exists to cap the bad-luck tail without paying it out to everyone. A flat rate would need a much higher base chance to produce the same average, which would hand early 5-stars to far more players. By leaving the first 73 pulls at the published base rate and then accelerating, the design keeps the advertised rate honest while still promising that nobody waits past 90.

The cost is that the last few pulls feel very different from the first few. At pull 88 the chance is 84.6% and at 89 it is 96.6%, so a player who reaches that point is almost certain to finish on the next pull. The bar chart on the results panel makes this visible: a long flat run, then a wall.

Because the numbers are not published, the exact shape is a matter of statistical inference rather than a rule you can read. If a game publishes its own curve, or if data from a large sample suggests a different step, change the inputs. The calculator is a model of whatever numbers you give it, and it says so on the page.

The 0.6% is not an official figure

The official statement is a consolidated 1.6%. Base 0.6%, plus six points per pull from the 74th, guaranteed by the 90th — that model comes from long-run player statistics, not from published rules, and the page says so.

The three numbers check each other: with the base rate of 0.6% held flat, the chance of one drop within ninety pulls is only about 42%. What lifts the consolidated figure to 1.6% is exactly the acceleration in the soft-pity stretch.

Soft pity starts at pull 74 and adds six percentage points per pull: 6% at 74, 42% at 80, 96% at 89 and 100% at 90. That curve is the explanation for why the last twenty pulls feel completely different.

Expectation and median have to be reported separately. The expectation is stretched by the rare cases that only drop on the final pull, while the median is the point by which half of players have already received one. Reporting expectation alone sets the wrong expectation about luck.

So the page states plainly that this is a community model. Dressing an estimate as official data is the easiest mistake for a tool like this to make and among the most damaging, especially when readers use it to decide whether to spend.