Gacha Pity Pulls
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A pity system guarantees a rare item after a certain number of unsuccessful pulls, overriding the base random probability once that threshold is reached, which caps the worst-case bad luck a player can experience.

Without pity, expected pulls is simply 1 divided by the drop rate; with pity, the expected number is always less than or equal to that pure-probability figure, since pity guarantees a success by the cap even in the unlucky tail of outcomes.

Hard pity is a fixed pull count where the rare item is guaranteed outright; soft pity is an earlier point where the drop rate itself starts increasing with each additional pull, making the guarantee point a true maximum rather than the typical outcome.

In games where pity counters carry over, your existing progress toward pity reduces the number of additional pulls needed for the next guaranteed item, so factor in your current pity count rather than assuming a fresh calculation from zero.

Pity only guarantees a maximum number of pulls to get a success — it doesn't prevent variance below that cap, so one player might get lucky early via the base random rate while another needs every pull up to the pity threshold, and both outcomes are normal under the same system.