Say a boss has a 1% chance to drop a rare item. Many players read that as "about one drop every 100 kills". It is not a promise, and it is not a schedule. It means that each kill, on its own, has a 1 in 100 chance, and that is all the game remembers about it.
Games use a random number generator (RNG): a bit of code that produces numbers that look random. For a 1% drop, the game picks a number from 1 to 100 and gives you the item if it is 1. Each roll is independent: the result of the last roll has no effect on the next. Ninety-nine misses in a row do not make roll 100 any more likely to win. Thinking they do is called the gambler's fallacy.
If each try has chance p and you try N times, the chance of getting at least one drop is 1 − (1 − p)N. The trick is to work out the chance of missing every time, then flip it. For p = 1%:
| Tries | Got at least one | Still empty-handed |
|---|---|---|
| 100 | about 63% | about 37% |
| 200 | about 87% | about 13% |
| 300 | about 95% | about 5% |
So after the "expected" 100 tries, more than one player in three still has nothing. Meanwhile about 1 in 100 players gets the drop on the very first try. Both are normal. Randomness comes in clumps and streaks; a perfectly even spread would be the suspicious thing.
In this simulator, one run is one player who keeps trying until the item drops. The bars show how many tries each run needed. Press the buttons to add runs and watch the shape appear: lots of quick lucky wins, and a long tail of players who were very unlucky.
Because the unlucky tail is so long, many games add pity (also called bad-luck protection). A counter tracks your misses, and once it hits a limit the next try is guaranteed to succeed. Some games instead use a "soft" version where the chance creeps up the longer you go without a drop. Turn pity on in the simulator and the tail gets chopped off: the worst case is now fixed, and the average number of tries drops a little. For a 1% rate with a guarantee at try 100, the average falls from 100 to about 63.
Notice what pity does not do. Inside the counter, each try is still a plain roll. It just sets a ceiling on how unlucky you can get.
Expected value is the average result over a huge number of tries. For a 1% drop it is 1 drop per 100 tries, or 100 tries per drop. That is a good number for comparing options (a 2% item is, on average, twice as cheap in tries as a 1% one) and a poor number for predicting your next hundred tries. If tries cost real money or a limited currency, budget for the long tail: to be 95% sure of getting a 1% drop without pity you need about 300 tries, three times the average.