Gacha rates

Gacha rates calculations on free banners

When pulling on a free banner, what are gacha chances? Calculating this may help you decide whether to pull, and on which banner. A Gallery of Dreams 10-Allies Bundle for 1,000 free CS guarantees at least one 4* gacha character, and have a 28.7% (1 in ~3,5) chance of pulling a random 5* gacha character. For detailed rates of specific gacha character look at “More details here” button in the Gallery of Dreams for the Encounter ratio table.

Calculation formula for rate of getting at least 1 result from several tries:

a is rate, b is 10th pull rate, n is amount of 10-pulls.

• 1-((1-a)^(9*n)*(1-b)^n)
• a=0.012, b=0.02, n=1; 1-((1-a)^(9*n)*(1-b)^n) = 12%
• a=0.012, b=0.02, n=5; 1-((1-a)^(9*n)*(1-b)^n) = 47%

Explanation of math above - how to calculate chance to get targeted characters:

• All chances of any desirable character should be added to get total rate total%/100% =(a)
  • If several characters are targeted, percentages of all such characters should be added.
  • Percentages of 4* forms of 5* characters should be also added if upgrading is considered.
• Calculate (1 - total rate) = rate of loss (1-a) for a single door
  • Same calculations for 10th door. Total rate (b) and rate of loss (1-b). If rate is different.
• Multiply all doors losses to get rate of cumulative loss.
  • (1-a)^9    *(1-b) is for 9 doors (1-a) with 10th door (1-b)
  • (1-a)^(9*n)*(1-b)^n is the same but for n times more 10-pulls.
• Chance to get result at least once = (1 - rate of cumulative loss) * 100%

For example, on the Regular Two Knights and the Holy Sword banner, if only Anabel NS 5* is targeted, it is 14,9% chance, or around 1 in 6.7, of getting her with a single 10-pull. If desirable result is either Anabel or Isuka either as 4* or 5*, it results in 24,8% chance, around 1 in 4.

But all calculations like these should be taken with precaution. Statistical math becomes relevant on much larger amount of repeats than even few tens of 10-pulls, that may be possible for a single player. Every single door is independent from all previous doors. Which means that even when calculation for as example 1.6% chance may show close to 100% result on 30x 10-pull, each next door will still have exact 1.6% to win and 98.4% chance to lose.