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