1. Base rate vs actual expectation
A 0.6% base rate does not mean "1 drop every 167 pulls". In the HoYoverse-style system with soft pity from pulls 74–89 (the per-pull rate jumps ~+6% each pull), the true expected value is closer to ~62 pulls.
- Genshin / Star Rail / Zenless: +6%/pull starting at 74, guaranteed at 90.
- Wuthering Waves: soft pity from 66, hard pity at 80.
- Arknights: +2%/pull starting at 50, guaranteed at 99.
2. The 50/50 trap
On HoYoverse featured banners, landing a 5★ resolves to 50% featured, 50% standard. Losing it guarantees the next 5★ is featured. Expected pulls for the first featured copy:
- Win the 50/50 (50%): ~62 pulls.
- Lose, then pull again (50%): 62 + 62 = 124 pulls.
- Expected value: ~93 pulls. Budget for roughly 2 5★s per featured character.
3. Worst-case (P90, P99)
The mean is misleading. The 90th-percentile worst case is often 1.5× the mean.
- Genshin 1 featured, median: ~90 pulls.
- Genshin 1 featured, P90: ~170 pulls.
- Genshin C6 (7 copies) median: ~720 pulls, P90: ~1,000.
Budget to P90 and you only fall short 1 banner in 10.
4. Why Monte Carlo matters
Soft pity + 50/50 + carried-over pity counters combine into a joint distribution that is painful to solve in closed form. 10,000 random runs gives an accurate empirical distribution.
- Single featured pull: closed-form math is enough.
- Multi-copy (eidolons / constellations) + 50/50 carry: simulation wins.
- Probability of success within your budget: read it straight off the histogram.
5. Currency conversion tips
- Genshin: 1 pull = 160 Primogems. 90 pulls ≈ 14,400 gems ≈ 6 months Welkin + 2–3 weeks free income.
- Star Rail: 1 pull = 160 Stellar Jade. Same scale.
- Blue Archive: 1 pull = 120 Pyroxene. 200 pulls = 24,000 pyroxene.
- Arknights: 1 pull = 600 Orundum. 99 pulls = 59,400 Orundum.
6. Pull or skip?
Three rules of thumb.
- You have ≥ P90 budget? → Pull.
- Meta-defining unit? → Pull at P50–P70 if you can absorb the variance.
- Just like the design? → Save until 1.5× budget accumulates.
7. Legal basis
Since March 2024, the Korean Game Industry Act mandates public disclosure of gacha rates. Every game's official probability page is linked above. This tool is a fan calculator over that public data; official pages always take precedence.