How to Calculate Gacha Odds: How Many Pulls Until You Get It, Pity, 50/50 and Max Copies

The short answer: the chance in n pulls is 1 − (1 − drop rate)^n

The chance of getting at least one copy of the unit you want in n pulls is found by subtracting the chance of getting none from 1. The chance of missing on a single pull is (1 − drop rate), so the chance of missing n times in a row is that number multiplied by itself n times: (1 − drop rate)^n. In other words, chance of getting it = 1 − (1 − drop rate)^n. At a 1% drop rate, the chance of missing once is 0.99, the chance of missing 100 times in a row is 0.99 to the 100th power, about 36.6%, and so the chance of getting it is about 63.4%.

The key point is that “1% = once in 100 pulls” describes the average (expected value); it does not mean “100 pulls guarantees it”. Even with an average of 100, a lucky player gets it on the first pull, and about 5% of players still have nothing after 300 pulls. How you account for that spread is the most important part of budgeting for gacha. The sections below explain how the numbers are calculated and how to read them.

What the formula means, and why 1% in 100 pulls is 63%

The basic assumption is that every pull is drawn with the same probability regardless of earlier results (each pull is independent). Under that assumption, the number of hits in n pulls follows a binomial distribution, and the chance of at least one copy comes straight from the formula above.

1. 1

   Convert the drop rate to a decimal

   1% becomes 0.01, 0.6% becomes 0.006 and 3% becomes 0.03. Working with percentages directly makes it easy to slip a decimal place, so convert first.

2. 2

   Find the chance of missing once

   That is 1 − drop rate: 0.99 for 1%, 0.994 for 0.6%.

3. 3

   Raise it to the number of pulls

   0.99 multiplied 100 times is about 0.366. On a scientific calculator use the x^y key; in a spreadsheet, type =0.99^100.

4. 4

   Subtract from 1

   1 − 0.366 = 0.634. The chance of at least one copy in 100 pulls at 1% is about 63.4%.

It feels like 1% times 100 pulls should be 100% because we tend to add probabilities. Adding 1% one hundred times does give 100%, but that counts overlaps, such as a player who got it on pull 1 getting it again on pull 2. A player who already has the unit doesn't add to the number of players with at least one copy. So more pulls push the chance closer to 100%, but never all the way.

The average (expected value) is also different from the number of pulls by which half of players have it (the median). At 1%, the average number of pulls until you get the unit is 100, yet half of players have it by pull 69. The average is pulled up by the minority who need 200 or 300 pulls. So “the average is 100, so 100 pulls gives better-than-even odds” is correct, but “100 pulls is safe” is not.

Drop rate × pulls table, and pulls needed for 50%, 90% and 99%

For a simple gacha with no pity or guarantees, this table shows the chance of at least one copy of the unit you want, for common drop rates and pull counts.

The next table turns it around: how many pulls until a given share of players has the unit. This is usually the more useful one when budgeting.

At every drop rate, the 90% point is about 2.3 times the average and the 99% point about 4.6 times the average. Remember “2.3× the average for 9 in 10 players, 4.6× for nearly everyone” and you can estimate without the table.

The chance of 2 or more copies, and the distribution of copies

If you are stacking duplicates to strengthen a unit (constellations, dupes, limit breaks), you need the chance of at least k copies rather than at least one. This is also a binomial calculation: add up the chances of exactly 0, exactly 1 … exactly k−1 copies and subtract from 1. The chance of exactly i copies is C(n, i) × p^i × (1 − p)^(n − i), where C(n, i) is the number of ways to choose which i of the n pulls are hits.

The average number of copies in 300 pulls is 3, yet the chance of 3 or more is only 57.8%, short of 60%. At the same 1% rate, half of players have 2 copies by pull 168, 3 copies by 268 and 7 copies (max) by 667. The more copies you want, the closer the requirement gets to “pulls for one copy × number of copies”, and the wider the spread becomes.

Odds and expected pulls with hard pity

Pity is a rule that guarantees a drop after a set number of pulls, such as “top rarity guaranteed on the 90th pull”. Most games reset the counter when a top-rarity item appears, and that's the type calculated here. Because pity caps the number of pulls, it also brings the average down a lot.

Notice that the chance by pull 89 is the same with or without pity. Pity doesn't raise the odds before it kicks in; it rescues the roughly 58.5% of players who still have nothing by pull 90. So in a 0.6% gacha with pity at 90, more than half of players end up pulling all the way to pity. The average is 69.7 because it blends the roughly 40% who got lucky early with the roughly 60% who got it exactly on pull 90.

Calculating the 50/50 and the guarantee after a loss

On a rate-up (limited) banner, a top-rarity pull is not necessarily the featured unit. Getting a top-rarity item other than the one you want is called losing the 50/50. A common rule is: “When a top-rarity item appears, there is a 50% chance it is the featured unit. If you lose, the next top-rarity item is guaranteed to be the featured unit.” This guaranteed state is often called the guarantee or hard guarantee.

The average of about 104.5 for the first copy is the average pulls until a top-rarity item (about 69.7) multiplied by 1.5, to cover the extra round when you lose the 50/50. The worst case is 180: nothing until pity at 90, lose the 50/50, then nothing until pity again, which is now guaranteed. Knowing this worst case before you start sets a hard ceiling on how much premium currency or money you may need.

With a 0.7% top rarity, pity at 80 and a 50/50 with guarantee, one featured copy takes about 92.1 pulls on average and 160 at worst; two copies take about 184.2 on average and 320 at worst. Pity just 10 pulls earlier cuts the worst case by 20 pulls per copy.

Carried-over pity and a carried-over guarantee

In games where the pity counter and the guarantee carry over to the next banner, how many more pulls you need depends heavily on where your counter is and whether you have a guarantee. For a 0.6% top rarity, pity at 90 and a 50/50 with guarantee, here is how many more pulls it takes to get one featured copy.

With 60 pulls on the counter and a guarantee, you get the featured unit within 30 more pulls even in the worst case. Starting from 0 without a guarantee, the worst case is 180. When deciding whether to pull now or save for the next banner, entering your counter and guarantee and comparing worst cases makes the choice much easier.

Pulls needed for max copies (7)

Going for 7 copies of the featured unit (max constellation or max dupes) makes the required number of pulls jump. Here are the pulls needed for 7 copies under common combinations of rules.

At ¥300 per pull, maxing a unit under the 0.6% / pity 90 / 50/50 system costs about ¥220,000 on average, about ¥270,000 at the 9-in-10 point of 902 pulls, and about ¥380,000 in the worst case of 1,260 pulls. One copy costs about ¥31,000 on average and ¥54,000 at worst. A rough rule: maxing out costs about seven times as much as one copy.

When collecting many copies, good and bad luck on individual pulls tend to cancel out, so the average and the median (50%) end up almost equal, and the spread is smaller relative to the number of copies than for a single copy. If you're aiming for max copies, budgeting about 1.25 times the average for systems with pity and a 50/50, or about 1.4 to 1.5 times for exchange-only or no-pity systems, roughly covers the 9-in-10 point.

Calculating with an exchange (spark) system

Some games let you exchange points earned from pulling for the featured unit, as in “after 300 pulls, trade your points for the rate-up character”. Copies obtained by exchange count too, so even with bad luck you are guaranteed one copy every set number of pulls.

With an exchange at 300, the worst case for the first copy is 300 pulls. And if points aren't consumed when the featured unit drops naturally, at pull 300 you have whatever you pulled plus the exchanged copy. That's why the 90% point for two copies, 328, is only slightly above 300. If exchange points don't carry over to the next banner, stopping partway wastes them, so factor in whether you'll pull all the way to the exchange.

How many random pin badges or trading cards to complete a set

How many blind items, such as random pin badges, acrylic stands, trading cards or capsule toys, you need to buy to collect every design in a set is known in mathematics as the coupon collector's problem. If every design is equally likely, the average number needed to collect all m designs is m × (1 + 1/2 + 1/3 + … + 1/m).

The formula adds up how many items it takes, step by step, to get a design you don't have yet. The first design comes with your first item, but once you have 9 of 10, the last one only appears 1 time in 10, so it takes 10 items on average. Getting stuck on the last few designs is why completing a set is so hard.

Take a set of 10 random pin badges and 30 purchases: you collect about 9.6 designs on average, get about 20.4 duplicates, and have about a 62.9% chance of a full set. With 20 purchases it's about 8.8 designs and 11.2 duplicates, and only about a 21.5% chance of a full set. If you plan to trade with friends, estimating your duplicates in advance tells you how many you'll have to offer.

How much harder secret items make a full set

Random merchandise often includes a rarer secret (chase) design. Its odds are often not published, but assuming “10 regular designs plus 1 secret at 2%”, the numbers look like this.

Including the secret roughly doubles the average, and at the 9-in-10 point you need 2.5 times as many items. The chance of at least one secret is about 18.3% in 10 items, about 45.5% in 30 and still only about 63.6% in 50. If you want the secret, it's more realistic to consider products sold by the box with a fixed assortment, or trading and buying the single design.

Common misconceptions about gacha odds

• “100 pulls at 1% guarantees it”: Without pity it's about 63%. About 37% of players get nothing in 100 pulls.
• “I haven't had it in ages, so it's due”: Apart from the pity counter, earlier results don't affect the next pull. The pull after 50 misses is still 1%, just like the first.
• “I just got one, so it won't come again for a while”: Same thing. The pull right after a hit has the same odds; only the pity counter resets.
• “The average is 100, so 100 pulls is safe”: The average is enough for only a little over half of players. The 9-in-10 point is about 2.3 times the average.
• “10-pulls are luckier than singles”: The top-rarity rate is the same. Only a separate guaranteed slot on the 10-pull changes what comes out.
• “Everyone else got it, so I will too”: On social media, people who got lucky post more than people who didn't. Even with 100 pulls at 1%, about 366 out of 1,000 players get nothing.

A probability describes what happens across many players pulling under the same conditions. Your own result is just one point somewhere in that spread, and you can't know where in advance. That's exactly why it helps to work out not just the average but also the 9-in-10 and worst-case numbers, and to decide beforehand how much you're willing to spend.

How to set a budget

Even after calculating the odds, you won't know whether you'll get the unit until it happens. The approach least likely to leave regrets is to decide in numbers how much you'll spend before you pull, and set the number of pulls to match.

1. 1

   Check the system

   Top-rarity rate, featured rate, pity count, whether a loss guarantees the next one, the exchange count and whether pity carries over. Check the rates and details pages.

2. 2

   Enter your current state

   Your current pity count, whether you have a guarantee, and how many pulls' worth of exchange points you have. These change how many more pulls you need.

3. 3

   Get the average, 90% and worst-case pulls

   Don't stop at the average: find the pulls by which 9 in 10 players succeed and the worst case set by pity or exchange.

4. 4

   Multiply by the price per pull

   Pull counts can feel abstract, so convert them to money using the price per pull (or the cash value of the premium currency).

5. 5

   Set your limit first, and stop there

   If you can afford the worst case, go ahead. If you can afford the 90% point, accept the remaining 1-in-10 chance of missing. If you can only afford the average, pull knowing there's about an even chance of missing.

The advantage of a gacha with pity is that the worst case is fixed, which makes budgeting easier. With no pity, or with secret items in random merchandise, even the 99% point can be 4 to 5 times the average, so carrying on without a limit can blow far past your budget.

How to use the Gacha Probability Calculator

The Gacha Probability Calculator does all of the calculations above in your browser using exact probabilities, not random simulations. The numbers you enter are not sent to a server; everything is calculated on your device.

1. 1

   Enter the drop rate and pulls on the “Chance to get” tab

   Enter the drop rate of the target (%), the number of pulls and how many copies you want. You get the chance of at least one (or N) copies, the chance of getting none (1 in X players), the average copies and the distribution of copies. Enter a price per pull to see the pulls for half of players, 9 in 10 players and 99% as a cost too.

2. 2

   Enter the gacha system on the “Pity, 50/50 & copies” tab

   Choose a preset under “Pick a common system”, or enter the top-rarity rate, the featured (rate-up) rate, pity (top rarity guaranteed on pull N), exchange (trade for the featured unit every N pulls) and whether a loss guarantees the next one. Enter how many copies you want (up to max) and how many pulls you can afford to see the chance, average and worst case.

3. 3

   Enter your current pity count and guarantee

   Enter the pulls since your last top-rarity item, your current exchange points and whether your guarantee is active, and the calculator works out how many more pulls you need from where you are.

4. 4

   Enter the number of designs on the “Blind-box set completion” tab

   Enter the number of regular designs, the number of secret designs and their odds, and how many items you'll buy. You get the chance of a full set, the average items needed, the number of duplicates, the chance of getting your favourite, and what to expect for each number of items bought.

5. 5

   Compare everything on the “Probability table” tab

   See the drop rate × pulls table, pulls needed by drop rate and the set completion guide by number of designs at a glance. Use “Copy result” to copy the result as text, or “Copy URL with these settings” to share the same settings with friends.

For tools whose odds are out in the open, like dice and wheels, see dice probability. For deciding among many people and running a fair draw, see rock-paper-scissors odds and how to run a fair draw.

FAQ

What is the chance of getting a 1% unit in 100 pulls?

About 63.4%. Multiplying the 0.99 chance of missing 100 times gives about 0.366, and subtracting that from 1 gives the answer. If 1,000 players do 100 pulls, about 366 get nothing. Half of players have it by 69 pulls, 9 in 10 by 230 and 99% by 459.

How do you calculate gacha odds?

The chance of at least one copy in n pulls is 1 − (1 − drop rate)^n. Convert the rate to a decimal (0.01 for 1%), subtract it from 1, raise the result to the nth power to get the chance of missing every time, then subtract that from 1. The chance of two or more copies uses the binomial distribution. With pity or a 50/50, you need a calculation that tracks the state (pity count and guarantee) pull by pull.

With a 0.6% rate and pity at 90, how many pulls on average?

About 69.7 pulls on average until a top-rarity item. But the chance of getting one by pull 89 is only about 41.5%, so more than half of players get it from pity on pull 90. For the featured unit with a 50/50 and a guarantee after a loss, it's about 104.5 pulls on average and 180 at worst.

What is the expected number of pulls for the featured unit with a 50/50?

It's 1.5 times the average pulls until a top-rarity item: about 104.5 with 0.6% and pity at 90, about 92.1 with 0.7% and pity at 80. The worst case is two full pities: 180 and 160 respectively. If your guarantee is already active, it equals the average until a top-rarity item.

How many pulls does max copies (7) take?

With a 0.6% top rarity, pity at 90 and a 50/50 with guarantee: about 732 pulls on average, 902 for 9 in 10 players and 1,260 at worst. At ¥300 per pull that averages about ¥220,000. Different systems give different numbers, so calculate with your game's pity, 50/50 and exchange rules.

My pity carried over. How many more pulls do I need?

With a 0.6% top rarity, pity at 90 and a 50/50, a counter at 60 without a guarantee means about 62.4 more pulls on average and 120 at worst; with a guarantee, about 27.5 on average and 30 at worst. Your counter and guarantee make a big difference, so the most reliable approach is to calculate from your current state.

How many random pin badges do I need to complete a set of 10?

If every design is equally likely, about 29.3 on average. Half of buyers complete it by 27, 9 in 10 by 44 and 99% by 66. Buying 30 gets you about 9.6 designs and 20.4 duplicates on average. Secret designs push the number up further.

If I haven't had it for a long time, is it more likely next time?

Not apart from the pity counter. Each pull is drawn independently of earlier results, so the pull after 50 misses has the same odds as the first. In a gacha with pity, though, the closer you are to pity, the shorter the worst-case number of remaining pulls.

Are the odds different for a 10-pull and single pulls?

The top-rarity rate is the same. A 10-pull is calculated the same as ten single pulls. If the 10-pull has a slot like “at least one 4★ guaranteed”, that slot changes the lower rarities; whether it raises the top-rarity rate is something to check on the game's rates page.