How pity works, and what a guarantee really does to your odds

In gacha games and loot boxes, pity means a guaranteed drop: a promise that you will get a rare item after a set number of pulls if luck hasn't already given you one. Some games call it bad luck protection, a guarantee or a pity timer. It sounds like it changes the odds a lot. In most games it changes the worst case far more than the average, and it helps to know which is which before you spend.

The starting point: pulls without pity

Every pull with a fixed, published chance p is independent. The game does not remember your bad run unless it has a pity system. Two formulas cover almost everything:

• Chance of at least one drop in n pulls: 1 − (1 − p)^n
• Average number of pulls until the first drop: 1 ÷ p

Take The Elder Scrolls Online. ZeniMax discloses a 2% chance that a Crown Crate includes an Apex reward. The average is 1 ÷ 0.02 = 50 crates. But after 50 crates the chance you have seen one is only 1 − 0.98^50, about 63.6%. You need 35 crates for an even chance, 114 to be 90% sure and 228 to be 99% sure. ZeniMax says some rewards become more likely after each miss, but it does not publish which rewards or by how much, so none of that can be counted here.

The average and the typical experience are not the same thing. A long tail of unlucky players pulls the average up, and without pity that tail has no end.

What a hard guarantee changes

A hard pity of N means pull N is guaranteed if the item hasn't dropped on pulls 1 to N − 1, and the counter then resets. That cuts off the tail. The average becomes:

Average pulls with a guarantee on pull N = (1 − (1 − p)^N) ÷ p

It is the chance that you get the drop before or on the guarantee, divided by the rate.

Worked example: Apex Legends

EA publishes a 7.4% chance of at least one Legendary item in an Apex Pack and states that a player cannot open more than 30 Apex Packs without receiving a Legendary item. So p = 0.074 and N = 30.

• Without the guarantee: 1 ÷ 0.074 = 13.5 packs on average.
• With the guarantee: (1 − 0.926^30) ÷ 0.074 = 12.2 packs on average.

The guarantee saves about 1.3 packs on average. That is modest, and here is why: even without it, 90.0% of players would already have a Legendary within 30 packs. Only players who miss on all of their first 29 packs reach the guarantee, and that is 0.926^29, about 10.8% of players.

What the guarantee really does is set a hard ceiling. Without it, one player in ten would still be waiting after 30 packs, and some would wait for 60 or more. With it, nobody waits more than 30.

The chances at a few pack counts, with the same 7.4% rate:

Packs opened	Chance of at least one Legendary
1	7.4%
5	31.9%
10	53.6%
20	78.5%
29	89.2%
30	100% (guaranteed)

The step from 89.2% to 100% at pack 30 is the whole effect of pity in this game.

The rule of thumb

Pity matters most when the rate is low compared with the guarantee. If most players would reach the item before the guarantee anyway, as in Apex Legends, the guarantee moves the average only a little. If the guarantee comes early compared with 1 ÷ p, it does most of the work, and the average gets close to the guarantee itself.

League of Legends shows the second case. Its Exalted Skins banner has a 0.5% rate and a guarantee by the 80th draw, so the average falls from 200 draws to about 66.1. Naraka: Bladepoint's Aeon Treasure is similar: a 0.6% rate with a guarantee within every 100 opens gives an average of about 75.4.

Two checks before you rely on a pity system:

1. Does it carry over? Some counters reset when an event ends or a banner changes. EA's Apex rule excludes Collection Event Packs, for example.
2. Is the guarantee for any rare item, or the one you want? Apex guarantees a Legendary item, not a particular one. A specific item is usually far rarer than its rarity group.

When there is no pity at all

Many drops have no disclosed guarantee. Warframe's official drop tables give an Intact Axi A1 Relic a 2% chance of its rare item, the same 50 average as the ESO crate. Refining the relic lifts that to 4%, 6% and 10%, and the averages fall to 25, about 17 and 10. Refining doesn't add a guarantee, but raising the rate has a similar effect on the tail: at 10% you reach a 90% chance in 22 relics, against 114 at 2%.

Lost Ark shows how layers multiply. Amazon Games gives a Legendary – Rare Card Pack a 4% chance of becoming a Legendary pack (25 on average), and each Legendary pack then picks one of 19 cards at 5.26% each. A specific Legendary card from those packs is two rare events in a row.

Soft pity

Some games raise the rate as you approach the guarantee. This is called soft pity. It lowers the average further, but only the publisher's own disclosure can tell you by how much. None of the games above publishes a soft-pity schedule, so we don't model one. Treat any figure you see for soft pity that doesn't come from the publisher as a guess.

Try it with your own numbers

The drop-rate calculator takes a rate and an optional guarantee and shows the chance within any number of pulls, the average, and how many pulls you need for a 50% or 90% chance. Every game page links to it with that game's rate already filled in. To see what a pull costs in pounds, read our guide to expected value for paid random items. To compare every rate a publisher discloses, see the drop-rate index.

How we checked

Every rate here comes from the publisher's own page, listed under sources, and matches our game pages. We re-read all of them on 11 November 2026, and none had changed. We worked out every figure with the formulas above and rounded only at the end.