This week just learned and try to understand the gacha probability modelling in Genshin Impact to obtain a 5 star character without paying anything (free) and there are interensting takes so far:
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Basic probability per pull
- The base probability of obtaining a 5 star character is 0,6%, which is 0,006 in decimal format
- This per pull probability is the foundation for all further calculations
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Segment based probability
- To calculate the chance of getting at least one 5 star character in a block of 10 pulls, you need to calculate based on this formulation: P(at least one 5 star character) = 1 - (1 - 0.006)^10 = 5,9 %
- This formula was successfully applied to calculate segments 14-23 pulls (particurlary in this case is myself)
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Probability addition is a mistake calculation that commonly occurs
- I initially thought i could add segment probabilities (for example directly calculated between 5.9% + 5.9%), however i just learned this is not valid approach, but instead i should use: P(at least one 5 star character in 20 pulls) = 1 - (1 - 0,006)^20 = 11,5 %
- In addition to that, consolidated probability is calculated using the complementary method, such as: P(at least one success pull) = 1 - P(no success in all pull). This could be applies whether you’re pulling 10, 20, or 80 times
- Reusing previous probability outcomes inside other probability formulas (for example, using 0.9410 in exponentiation after 2, 3 times calculation) misrepresents the actual probability behavior
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Soft pity mechanism
- After around 75–76 pulls, the probability of getting a 5 star character increases and i just learned to estimate soft pity pulls with a higher per-pull rate, such as 15%. Using this, i tried to calculated for pulls at segments 83–92 (10 pulls at 15%) and got a calculation: P(at least one 5 star character) = 1 - (1 - 0,15)^10 = 80,4 %
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Final pull probability
- From pull 13 to 92 (total 80 pulls), the probability of myself is not getting a 5 star character is around 12-13 % based on this calculation: (1 - 0,006)^70 × (1 - 0,15)^10 = 12,8 %
- Thus, the chance of getting a 5 star character by pull 92 is: 87,2 % (1 - 0,128)
Next Steps?
- Try building a spreadsheet or small simulator to input my pity count and get real time odds
- Explore expected value to estimate how many primogems are needed for a likely 5 star character
- Explore and have a better understanding to model guaranteed pity and 50/50 win-loss mechanics