The published rate is per pull, and nobody makes one pull. The number you want is the chance across however many pulls you can afford, and the gap between the two is wide enough that most intuitions about it are wrong.
The formula
Pulls are independent, so the chance of missing every time is the miss chance multiplied by itself however many times. The chance of at least one success is therefore
1 − (1 − rate)^pulls
At a rate of 0.6 per cent, ten pulls gives 5.8 per cent. Not 6 per cent, which is what multiplying the rate by ten would suggest, but close enough that the difference does not yet matter. It matters later: fifty pulls gives 26 per cent rather than 30, and the gap keeps widening because you can never exceed certainty.
That 26 per cent is the number worth sitting with. Fifty pulls at a 0.6 per cent rate and you are still more likely than not to have nothing. The gacha pity calculator reports the chance of nothing alongside the chance of success for exactly this reason.
What pity actually does
A hard pity count changes the shape of the problem completely, and not in the way most explanations suggest.
Without pity, the expected number of pulls to your first copy at 0.6 per cent is one divided by 0.006, which is about 167. With a guarantee at pull 90, it is about 70. The guarantee does not raise your chance on any individual pull, it chops off the long tail of catastrophic runs, and the tail was carrying most of the average.
It also means the only number in the whole system that guarantees anything is the pity count. Saving to 90 removes the gamble entirely. Saving to 50 does not get you most of the way there, it gets you to roughly a one in four chance.
Dry streaks are the default
The chance of nothing in fifty pulls is 74 per cent. In eighty-nine pulls, one short of pity, it is 58 per cent. A player who reaches the guarantee without a single early success is not unlucky, they are in the majority.
This is the part that feels wrong, and the feeling has a name. Pulls have no memory. Twenty consecutive failures do not make the twenty-first more likely, because the coin does not know what it did before. The only thing that genuinely improves with each failure is your proximity to the pity counter, which is a rule in the game rather than a property of probability.
Two things the arithmetic does not cover
Soft pity, where the rate ramps up over the last stretch before the guarantee, makes real systems kinder than the flat calculation suggests. Treat the flat number as a floor.
The other is the coin flip on limited banners, where winning the pull still leaves you a fifty-fifty chance of the character you wanted. If your game has that, halve your effective rate or double the pulls you budget, because the published rate is not the rate of getting what you came for.
For the same arithmetic applied to farming an item from an enemy, there is the drop rate calculator, and for the other long grind there is the piece on levelling curves.