A negative binomial approach to drop rates

Qleak

A negative binomial distribution can be used to estimate the number of trials needed before having a fixed number of successes. Don't eat your hat and set it up in the alternative fashion (counting the number of successes before a fixed number of failures) like i did in this discussion
https://forums.galaxy-of-heroes.starwars.ea.com/discussion/21257/rey-drop-rate-32-72/p1


Character Shards
Let's start with shards you need 330 shards to 7* a character and all data points to the probability of success is p=1/3. The average number of trials needed before success will be 990. The distribution looks like this:

p=1/3 r=330



And percentile ranks look like this (I've also added in the energy costs)

%tile	tries	8 EN	10 EN	12 EN	16 EN	 20EN
-----------------------------------------------------
0.01 	891	7128	8910	10692	14256	17820
0.05 	919	7352	9190	11028	14704	18380
0.10 	935	7480	9350	11220	14960	18700
0.15 	945	7560	9450	11340	15120	18900
0.20 	953	7624	9530	11436	15248	19060
0.25 	961	7688	9610	11532	15376	19220
0.30 	967	7736	9670	11604	15472	19340
0.35 	973	7784	9730	11676	15568	19460
0.40 	979	7832	9790	11748	15664	19580
0.45 	985	7880	9850	11820	15760	19700
0.50 	990	7920	9900	11880	15840	19800
0.55 	996	7968	9960	11952	15936	19920
0.60 	1001	8008	10010	12012	16016	20020
0.65 	1007	8056	10070	12084	16112	20140
0.70 	1014	8112	10140	12168	16224	20280
0.75 	1021	8168	10210	12252	16336	20420
0.80 	1028	8224	10280	12336	16448	20560
0.85 	1037	8296	10370	12444	16592	20740
0.90 	1049	8392	10490	12588	16784	20980
0.95 	1066	8528	10660	12792	17056	21320
0.99 	1098	8784	10980	13176	17568	21960

Note you'll only 7* a character within 990 tries half the time, so the other half it will take longer. But only 1 time in 100 will you expect it to take longer than 1098 tries.

Gear
For the purple gear, I don't have as good of data as character shards. From what I understand the drop rate is somewhere around 1/5 or 1/6. For this example I assume p=1/5 and assume we are looking for 20 purple whatevers

p=1/5 r=20

%tile	tries	6 EN	8 EN	10 EN
---------------------------------------
0.01	61	366	488	610
0.25	87	522	696	870
0.50	100	600	800	1000
0.75	114	684	912	1140
0.99	154	924	1232	1540

Here's what it looks like to farm 50 purple pains
p=1/5 r=50

%tile	tries	6 EN	8 EN	10 EN
---------------------------------------
0.01	184	1104	1472	1840
0.25	229	1374	1832	2290
0.50	250	1500	2000	2500
0.75	271	1626	2168	2710
0.99	331	1986	2648	3310

Anyhow, that's more than enough for now. Are there any particular p and r values you want to see. Let me know below.

Leapster

I love stats. Thanks for doing this and sharing it. From my experiences this reflects the results very well.

Qleak

People usually complain that the last few shards the drop rate seems to decrease perhaps what they are noticing is more skew to the distribution

p=1/3 r=3

%tile	tries
-------------
0.01 	4
0.25 	7
0.50 	9
0.75 	12
0.99 	23

Here's the last 3 purple pains
p=1/5 r=3

%tile	tries
-------------
0.01 	5
0.25 	10
0.50 	15
0.75 	20
0.99 	40