Oh hey. Math. I'm really freaking bored, so why not.

...

Pulling up the odds for a single roll in the "Winner's Gacha", they are as follows:

* 0.5% : Six star
* 6% : Five star
* 30% : Four star
* 63.5% : Three star

Therefore, the odds to get "at least an X star" in a single roll are:

* 0.5% : At least a six star (duh)
* 6.5% : At least a five star
* 36.5% : At least a four star
* 100% : At least a three star (more duh)

...

Now, for multiple rolls. Let's find out the odds of getting N number of "five stars or greater" in a 10+1 roll. Really, it boils down to the same math involved in "getting heads N times total in 11 flips", just with a probability of heads that's not 50%.

* Probability of success (a five star or above): Ps = 6.5%
* Probability of failure (a four star or below): Pf = 100-6.5 = 93.5%

The probability of 11 failures - that is, 11 pulls and everything is four stars or below - is quite simple to find. After all, there's only one possible result.

* Probability of 11 failures = Pf^11 = 0.935^11 = 0.4774 = 47.74%

The probability of 1 success and 10 failures is only slightly trickier. We 1) find all possible ways we could hit 1 success/10 failure, 2) calculate the probability of each, and 3) sum them up. There are 11 ways (one for each of the positions), and each of those ways has identical odds (since they're independent), so it's not that tough.

* Probability of 1 success and 10 failures = 11*(Ps*(Pf^10)) = 11*(0.065*(0.935^10)) = 0.3651 = 36.51%

Going further up the chain gets more taxing to do by hand, as the number of permutations grows (well, going further towards the middle does; the far end is just as simple to calculate as the near one). But the same basic math still applies. And, thankfully, computers are damned good at doing simple math exceedingly fast.

Probability of getting exactly N shinies (five star or greater) in a 10+1 pull
--------------------------------------------------------------------
* 0 : 47.74498042254755603 %
* 1 : 36.51086738194813819 %
* 2 : 12.69094320762902939 %
* 3 : 2.64677425185846049 %
* 4 : 0.36800069811935815 %
* 5 : 0.03581611072605517 %
* 6 : 0.00248989005047442 %
* 7 : 0.00012363854337726 %
* 8 : 0.00000429759642755 %
* 9 : 0.00000009958779600 %
* 10 : 0.00000000138464315 %
* 11 : 0.00000000000875078 %

(And for anyone concerned, the sum of those numbers - not the shown numbers, the actual ones used in the underlying script's array = totals to "1.0000000000000004". So there's a tiny bit of precision error in there, past 15 decimal places or so. But that's still good enough for a few significant figures, even for the smallest probability above. Besides, even if the 11 success case was 100x larger, it's still really damned microscopic.)

...

You could derive other interesting numbers - for example, the total probability of getting 4 or greater shinies in a single 10+1 is actually slightly rarer (0.41%) than getting a single six star in a single pull (0.5%) - but meh.

That was fking awesome. I love math too.

Well I just had my 12.69% win with 2 golds from 1 10+1 pull. Madonna (new for me and now my 5th 5* skill activation girl!) and Nasturtium (A dupe but the 1st I have gotten of her so at least an extra equip slot).

Congrats on both those golds and beating the 12.69% odds, Zandel. Madonna's waifu tier up there with Anthurium and dupes are always a welcome addition.

Just to test I put 5 1.2xskill girls on one team... it's a 2x total but if you swap one with Cymbidium it becomes 3.6x round 1 and 1.8x after... so enough to kill anything in a few rounds tops.

Originally Posted by Zandel

Just to test I put 5 1.2xskill girls on one team... it's a 2x total but if you swap one with Cymbidium it becomes 3.6x round 1 and 1.8x after... so enough to kill anything in a few rounds tops.

Interesting.
Well at least reading through the last few posts I finally got an Idea how that buff stacks.
I always though it to be weird that "Oh 5. 1,2 6x the activate rat, so if my girl has 30% activation rate, times 6 I have a guaranteed 100% of her activating it), or at least thats what I used to think. Granted I considered it to be kinda wonky and over the top.
So the scaling is much reduced, 0,2 per girl to be precise.
Still a 30% skill rate, time 1,8/ / 2,0 is over the 50/50 range which makes it pretty solid. It requires a full skill act. team though. But certainly is solid.

Originally Posted by Myrdin

Interesting.
Well at least reading through the last few posts I finally got an Idea how that buff stacks.
I always though it to be weird that "Oh 5. 1,2 6x the activate rat, so if my girl has 30% activation rate, times 6 I have a guaranteed 100% of her activating it), or at least thats what I used to think. Granted I considered it to be kinda wonky and over the top.
So the scaling is much reduced, 0,2 per girl to be precise.
Still a 30% skill rate, time 1,8/ / 2,0 is over the 50/50 range which makes it pretty solid. It requires a full skill act. team though. But certainly is solid.

I have noticed that the % activation rate for skills is not really accurate. I mean they say 20-30% on average but those skills seem to activate more like 40-50% of the time in reality. I have watched over hundreds of activations and even tho it COULD be an RNG streak it still seems higher than expected. I have also notices that 1.2x activation (a 20% buff so from 20 - 24 or 30 - 36) seems to have a much larger effect than the numbers suggest.

I'm thinking that it's the RNG generator they are using, since computers cannot ever do anything 100% random i'm going to guess that the algorithm they are using is slightly weighted towards the low (or high not sure what skills use) end.