Gacha Probability Calculator Guide: Pull Rates, Pity Systems & Expected Cost
Quick Answer
- *The formula for at-least-one success: P = 1 – (1 – r)^N, where r is the drop rate and N is the number of pulls.
- *At a 0.6% base rate (Genshin Impact), you need 115 pulls for a 50% chance without pity — but soft pity brings the average down to about 62.
- *Worst-case guarantee for a featured character in most gacha games costs $300–$400 USD.
- *The global gacha game market generated $23.4 billion in revenue in 2024 (Sensor Tower).
How Gacha Probability Works
Gacha systems are digital loot boxes where you spend currency (free or paid) for a random chance at characters, weapons, or items. The name comes from Japanese capsule toy machines (gashapon). The math behind them is straightforward probability theory, but the game design around them is engineered to feel more generous than it actually is.
Each pull is typically an independent event with a fixed probability. If a 5-star character has a 0.6% drop rate, that means each individual pull has a 0.6% chance of success — regardless of how many pulls you've already done (unless the game has a pity system).
The Core Probability Formula
The probability of getting at least one success in N independent pulls is:
P(at least one) = 1 – (1 – r)^N
Where:
- r = per-pull success rate (e.g., 0.006 for 0.6%)
- N = number of pulls
Probability Table (0.6% Base Rate, No Pity)
| Number of Pulls | Chance of At Least One 5-Star |
|---|---|
| 10 | 5.8% |
| 30 | 16.5% |
| 50 | 25.9% |
| 75 | 36.2% |
| 100 | 45.2% |
| 150 | 59.3% |
| 200 | 69.9% |
| 300 | 83.5% |
Without pity, you need 115 pulls for a coin flip (50%)and 383 pulls for a 90% chance. This is why pity systems exist — pure randomness is brutally unforgiving at low drop rates.
How Pity Systems Work
Most modern gacha games include pity mechanics to cap the worst-case scenario. There are two main types:
Hard Pity
A guaranteed drop at a specific pull count. In Genshin Impact, you are guaranteed a 5-star within 90 pulls. No exceptions. This counter resets every time you pull a 5-star. Japan's JOGA (Japan Online Game Association) guidelines and China's regulations effectively require games to disclose rates and implement some form of ceiling, which drove widespread pity adoption after 2017.
Soft Pity
The drop rate increases gradually before hard pity. In Genshin Impact, community data mining (from Paimon.moe and other trackers analyzing over 10 million pulls) shows that the 5-star rate jumps from 0.6% to approximately 6% per pull starting at pull 74. This means most players get their 5-star between pulls 74–80, not at pull 90.
Pity Comparison Across Games
| Game | Top Rarity Rate | Soft Pity Start | Hard Pity | Avg Pulls |
|---|---|---|---|---|
| Genshin Impact | 0.6% | 74 | 90 | ~62 |
| Honkai: Star Rail | 0.6% | 74 | 90 | ~62 |
| Wuthering Waves | 0.8% | 66 | 80 | ~55 |
| Zenless Zone Zero | 0.6% | 74 | 90 | ~62 |
| Arknights | 2.0% | 51 | None (99% by ~70) | ~35 |
| Fate/Grand Order | 1.0% | None | None (spark at 330) | ~100 |
The 50/50 and Guarantee System
Many games add a second layer: the 50/50. When you pull a 5-star in Genshin Impact, there's a 50% chance it's the featured character and a 50% chance it's a standard 5-star. If you "lose" the 50/50, your next 5-star is guaranteed to be the featured character.
This means the true worst case for a specific featured character is 180 pulls (lose 50/50 at pull 90, then hit pity again at pull 180). In practice, the average is about 105 pulls per featured character when accounting for soft pity and 50/50 odds.
Expected Cost Analysis
Gacha currency pricing varies, but here's a typical conversion using Genshin Impact's Genesis Crystal pricing (with first-time top-up bonus):
| Scenario | Pulls Needed | Approx. USD Cost |
|---|---|---|
| Lucky (early 5-star, win 50/50) | ~30 | ~$50–$60 |
| Average (soft pity, win 50/50) | ~62 | ~$110–$130 |
| Average (soft pity, lose 50/50) | ~124 | ~$220–$250 |
| Worst case (double hard pity) | 180 | ~$340–$370 |
According to Sensor Tower, the average gacha game player who spends money pays about $10–$25 per month. But the top 1% of spenders ("whales") account for approximately 50% of total gacha revenue, spending $500+ per month. A 2024 Newzoo report found that gacha mechanics generated $23.4 billion globally that year.
Common Probability Traps
The Gambler's Fallacy
In systems without pity, past results don't change future odds. Pulling 89 non-5-stars in a row does not mean pull 90 has better odds (unless there's a programmed pity system). Each pull is independent. The universe doesn't owe you luck.
Sunk Cost Trap
"I've already spent $100, I might as well keep going." The $100 is gone regardless. Whether to keep pulling should depend only on whether the remaining expected cost is worth the character to you right now. Past spending is irrelevant to that decision.
Rate Misperception
A 0.6% rate sounds small, but people consistently overestimate what it means in practice. At 0.6%, you have a 54% chance of NOT getting the item even after 100 pulls. Many players expect "around 50 pulls should do it" when the math says otherwise.
Calculate your exact gacha odds before you pull
Use our free Gacha Probability Calculator →Frequently Asked Questions
What are the odds of getting a 5-star in Genshin Impact?
The base rate for a 5-star character in Genshin Impact is 0.6% per pull. Soft pity kicks in at pull 74, dramatically increasing the rate to roughly 6% per pull. Hard pity guarantees a 5-star at pull 90. Due to soft pity, the average number of pulls for a 5-star is about 62, not 90. With the 50/50 system, you have a 50% chance of getting the featured character on any 5-star pull.
What does pity mean in gacha games?
Pity is a guaranteed drop mechanic that ensures you get a rare item after a certain number of unsuccessful pulls. For example, Genshin Impact guarantees a 5-star within 90 pulls. Some games have "soft pity" where the drop rate gradually increases before the hard pity ceiling. Pity counters typically reset when you pull the target rarity.
How much does it cost to guarantee a character in gacha games?
It varies by game. In Genshin Impact, worst case (losing 50/50 at hard pity twice) requires 180 pulls. At roughly $2 per pull using top-up bonuses, that's about $360 USD. In Honkai: Star Rail, the worst case for a limited 5-star is also 180 warps, costing approximately $340–$370. Average cost is much lower — typically $150–$200 due to soft pity.
How do you calculate the probability of getting a character in N pulls?
Without pity, use the formula: P(at least one) = 1 – (1 – r)^N, where r is the per-pull rate and N is the number of pulls. At a 0.6% rate, 100 pulls give you a 45.2% chance. With pity systems, the math is more complex because the rate changes at certain thresholds, requiring simulation or step-by-step probability calculations.
What is the gambler's fallacy in gacha?
The gambler's fallacy is the belief that past bad luck makes future good luck more likely. In gacha games without pity, each pull is independent — 100 failed pulls do not make the 101st more likely to succeed. However, games with pity systems do genuinely increase your odds over time, which is why pity exists. The fallacy only applies to pure random systems.