🎲 Summon Rate Calculator
Calculate Your Gacha Pull Probabilities and Expected Attempts
Calculate Summon Probability
Understanding Summon Rates in Gacha Games
Summon rate calculators are essential tools for players of gacha games who want to understand their chances of obtaining desired characters or items. Whether you're playing Genshin Impact, Honkai Star Rail, Fate/Grand Order, or any other gacha game, understanding probability mathematics can help you make informed decisions about your resources.
What Are Summon Rates?
Summon rates represent the probability of obtaining a specific character or item from a random pull in a gacha system. These rates are typically expressed as percentages, with rare characters having much lower rates than common ones. For example, a 5-star character might have a 0.6% base rate, meaning each individual pull has a 0.6% chance of being that character.
Understanding Probability Mathematics
The probability of getting at least one success in multiple pulls is calculated using the complement rule:
P(at least one success) = 1 – P(all failures)
For example, with a 0.6% (0.006) base rate over 90 pulls:
- Probability of failure on one pull = 1 – 0.006 = 0.994
- Probability of failure on all 90 pulls = 0.994^90 = 0.582
- Probability of at least one success = 1 – 0.582 = 0.418 or 41.8%
Types of Pity Systems
Hard Pity
Hard pity guarantees a successful pull after a specific number of attempts. In Genshin Impact, for example, you're guaranteed a 5-star character within 90 pulls. This ceiling ensures players won't experience infinitely bad luck.
Soft Pity
Soft pity gradually increases the summon rate as you approach the hard pity threshold. In many games, this increase starts around pull 74-76 and dramatically boosts your chances. The rate might jump from 0.6% to over 30% per pull in the soft pity range.
No Pity
Some older gacha games operate without pity systems, relying purely on statistical probability. These systems can lead to extreme variance in player experiences.
Example Calculation: Genshin Impact 5-Star Character
Scenario: You want to calculate your chances of getting a specific 5-star character.
- Base Rate: 0.6%
- Number of Pulls: 90 (reaching hard pity)
- Pity System: Hard Pity with Soft Pity
- Pity Threshold: 90 pulls (hard), 74 pulls (soft starts)
- Soft Pity Rate: 32.4% (at pull 89)
Result: With soft pity mechanics, your cumulative probability reaches nearly 100% by pull 90, with most 5-stars appearing between pulls 76-85.
Expected Value and Average Pulls
The expected value tells you the average number of pulls needed to get your desired result. For a simple base rate without pity:
Expected Pulls = 1 / (Base Rate as decimal)
For a 0.6% rate: 1 / 0.006 = 166.67 pulls on average. However, pity systems significantly reduce this number in practice.
Featured vs. Standard Banners
Most gacha games distinguish between:
- Featured Banners: Higher rates or guaranteed featured characters after losing 50/50
- Standard Banners: Permanent pools with diluted rates across all characters
- Rate-Up Events: Temporary increases to specific character rates
The 50/50 System
Many games implement a 50/50 system where your first 5-star has a 50% chance of being the featured character. If you lose the 50/50 and get a standard character, your next 5-star is guaranteed to be the featured one. This affects your expected resource investment significantly.
Calculating Resource Requirements
Once you know the probability, you can calculate resource needs:
- Premium Currency per Pull: Typically 160 primogems in Genshin Impact
- Multi-Pull Discounts: 10-pulls might cost 1600 instead of 1600
- Free Resources: Daily commissions, events, and login bonuses
- Worst-Case Scenario: 180 pulls × 160 = 28,800 primogems for guaranteed featured character
Statistical Variance and Luck
Understanding variance is crucial for managing expectations:
- About 50% of players will need fewer than average pulls
- About 50% of players will need more than average pulls
- Extreme luck (positive or negative) is statistically rare but possible
- Pity systems reduce variance and protect against worst-case scenarios
Practical Applications
Resource Planning
Use the calculator to determine if you have enough saved resources for upcoming banners. If you want a character with 90% confidence and currently have 50 pulls saved, you can calculate whether you need to purchase additional currency.
Banner Selection
When multiple attractive banners are available, probability calculations help you prioritize which characters you can realistically obtain with your resources.
Long-Term Strategy
By understanding expected values, you can plan months ahead for highly desired characters, ensuring you accumulate sufficient resources through free play.
Example: Planning for Limited Character
Goal: Obtain a limited 5-star character with 95% confidence
Current Progress: 0 pity, no guarantee
Calculation:
- To reach 95% probability with soft pity included: approximately 85-88 pulls
- With 50/50 risk: may need up to 170 pulls for 95% confidence of featured character
- Resource requirement: 170 × 160 = 27,200 primogems
- Time to accumulate free primogems: approximately 4-5 months of active play
Advanced Considerations
Constellation/Dupe Systems
Some games reward duplicate characters with power increases. Calculate the expected pulls for C6 (6 duplicates) by multiplying the single-character expectation by 7, accounting for pity resets.
Weapon Banners
Weapon banners often have different pity systems. Genshin Impact's weapon banner has a fate point system requiring up to 240 pulls for a guaranteed specific weapon.
Rate-Up Mechanics
During rate-up events, the base rate might increase from 0.6% to 1.2%, significantly improving your odds and reducing expected pulls from 167 to 83.
Common Misconceptions
- Gambler's Fallacy: Previous failures don't increase your next pull's base rate (without pity)
- Guaranteed Confusion: Hard pity guarantees a 5-star, but not necessarily the featured one (50/50)
- Probability Stacking: Two separate 50% chances don't equal 100%; they equal 75%
- Small Sample Bias: Your personal experience may differ significantly from statistical expectations
Tips for Responsible Gaming
- Set strict budgets before summoning and stick to them
- Never chase losses or try to "make up" for bad luck
- Remember that gacha systems are designed to encourage spending
- Focus on enjoying the game with characters you already have
- Use calculators to make informed decisions, not to justify excessive spending
Comparing Different Games
Different gacha games have vastly different generosity levels:
- Genshin Impact: 0.6% base, 90-pull hard pity, soft pity starting at 74
- Honkai Star Rail: 0.6% base, 90-pull hard pity, similar soft pity mechanics
- Fate/Grand Order: 1% base rate for SSR, no pity system (JP version)
- Arknights: 2% base for 6-star, pity increases by 2% per pull after 50
Using This Calculator Effectively
To get the most accurate results:
- Look up your game's exact summon rates (usually in-game information or official sources)
- Identify the pity system type and threshold
- Input your current pity count and number of intended pulls
- Review cumulative probability and expected pulls
- Calculate worst-case scenario resources needed
- Make informed decisions about summoning or saving
Conclusion
Understanding summon rate mathematics empowers you to make better decisions in gacha games. While luck plays a significant role, probability calculations provide realistic expectations and help prevent disappointment. Use this calculator to plan your summons strategically, manage your resources effectively, and enjoy gacha games responsibly without overspending or developing unrealistic expectations.