Number of pulls needed to cover all 3 Latest 5*s

justsing

With the addition of 5* covers in boss events, I thought it'd be nice to run some simulations to see how many pulls are now needed to finish all 3 Latest 5*s.

All scenarios were done with 100,000 simulation runs.

Assumptions:

1. Boss events start you out at 1/1/0 for each 5*.
2. Progression shards from SCL10 PVE give you 2 covers for each 5*. Those covers are applied at the end, meaning you only need 11 usable covers from boss event + Latest pulls.
3. Not factoring in champ rewards, feeders, placement shards, or converting saved covers.
   

Scenarios:

1. Boss + SCL10 PVE - start at 1/1/0 from boss event + 1000 shards from PVE progression
2. SCL10 PVE only - 1000 shards from PVE progression
3. Boss covers only - start at 1/1/0 from boss event
4. None - start at 0/0/0 for each 5*

Results

Scenario	Mean (SD)	Median (P25, P75)	P5, P95
Boss + SCL10 PVE	252 (61)	244 (209, 286)	168, 363
SCL10 PVE only	300 (66)	292 (254, 338)	208, 419
Boss covers only	342 (86)	329 (282, 387)	228, 500
None	390 (90)	377 (327, 438)	269, 556

Notes:

• SD = standard deviation, measures the amount of variation in the data
  
• P25 = 25th percentile, 25% of the 100k simulation runs needed at most <P25> pulls to cover all 3 Latest 5*s
• P75 = 75th percentile, 75% of the 100k simulation runs needed at most <P75> pulls to cover all 3 Latest 5*s
• P5 = 5th percentile, 5% of the 100k simulation runs needed at most <P5> pulls to cover all 3 Latest 5*s
• P95 = 95th percentile, 95% of the 100k simulation runs needed at most <P95> pulls to cover all 3 Latest 5*s
• To reduce clutter, plot does not include SCL10 PVE only scenario
  

Discussion

The 300 pulls estimate was from a time where CS did 1:1 cover swaps for Latest 5*s. Once the cover swaps went away, that estimate became closer to 400 pulls.

Adding 5* shards to PVE helped a lot. With full progression in SCL10 PVE, a player can get at least 2 covers worth of shards for a Latest 5*. With those progression shards, the number of pulls needed dropped back down to 300 pulls.

Now with the addition of 2 covers from boss event progression, the number of pulls needed drop by another 50 pulls on average. Therefore, if you're getting 2 covers from boss events and getting all progression shards from SCL10 PVE, you only need ~250 pulls to cover all three Latest 5*s.

Keep in mind these numbers do not account for other factors, such as placement shards, shards from favoriting, or champ rewards. So it isn't unlikely that you'd need fewer than 250 pulls, but it is a decent ballpark estimate.

Note: There was a bug in the earlier version of the simulation code. It has since been updated/rerun, and numbers have been crosschecked with another player's simulation results (many many thanks to unobtainium404).

Vhailorx

Am I reading this wrong, or does 224 pulls give a 75% chance of covering all 3 (or better)?  Seems like somewhere between 200 and 225 should be the new target point.

Bowgentle

Oof at that possible 380 pulls without any additional covers.

justsing

Apologies to all! There was an error in my code/results when I first posted. It has since been corrected, and the results have been crosschecked with another player's simulation results (many many thanks to unobtainium404).

bahamut685

Commend the efforts, but since we're never going to get any information on standard deviations [for pull rates], it only really represents the expected values, if the pulls you do are in the magical sweet spot, where your pull rate matches the official pull rates, and your spread of covers is remotely sane (I'm sure that at least half of us have hit a ridiculous point of 5/2/0 with 7 saved covers).

TheEyeDoctorsWife

This mirrors my current pull data but could entirely be coincidental. Barring the aforementioned design flaws. Nice work 

justsing

bahamut685 said:

Commend the efforts, but since we're never going to get any information on standard deviations [for pull rates], it only really represents the expected values, if the pulls you do are in the magical sweet spot, where your pull rate matches the official pull rates, and your spread of covers is remotely sane (I'm sure that at least half of us have hit a ridiculous point of 5/2/0 with 7 saved covers).

My intent was to give people a ballpark number, so that they know how big of a hoard they need. If you're concerned about bad pull rates and want a more conservative estimate, that's what the 75th and 95th percentile estimates are for.

trenchdigger

Just to be 100% clear that I am interpreting this data correctly:

Using the boss + SCL10 PVE scenario, if I always pull 286 times for each set of 3 latest, then 3 times out of 4 I will cover all 3, but 1 time out of 4 I will not?

And even if every time I pull 363 times, then 19 times out of 20 I will cover all 3, but 1 time out of 20 I will not.

justsing

trenchdigger said:

Just to be 100% clear that I am interpreting this data correctly:

Using the boss + SCL10 PVE scenario, if I always pull 286 times for each set of 3 latest, then 3 times out of 4 I will cover all 3, but 1 time out of 4 I will not?

And even if every time I pull 363 times, then 19 times out of 20 I will cover all 3, but 1 time out of 20 I will not.

It means 75% of the people who pull will likely cover all 3 in 286 pulls. Doesn’t mean that it’ll happen to you 3 out of 4 times. 

trenchdigger

Yes understood, if I have average luck then I should expect to cover all 3, 3 times out of 4 if I pull 286 times.

jsmjsmjsm00

Did this account for undesired cover distributions or were you only considering getting to 13 covers per 5*?

Chirus

jsmjsmjsm00 said:

Did this account for undesired cover distributions or were you only considering getting to 13 covers per 5*?

I think you'll want to look for the P values here. They will account for uneven cover distributions. P95 is the point where you're highly unlikely to still be screwed from poor luck, that is, 95% of the sample population will have already fully covered all 3 latest legends.

jsmjsmjsm00

That doesn't answer my question. 

Is this calculation considering that there are three different covers per character or simply that you need to hit 13 pulls for a character?

To clarify, are the statistical  distributions shown just the number of pulls to reach 13 covers per character, therefore including un-champable cover distributions? If so, then these would be fairly significant underestimates of how many pulls people should hoard for.

jsmjsmjsm00

I've run a quick calc that doesn't account for cover distribution and hit a mean of 320ish pulls to cover the three 5s with no other shards or covers. I'm going to assume then, since this post presents a mean higher than that (390), that it is accounting for cover distribution.

EDIT: Ran my own simulation to confirm the case accounting for cover distribution and got 390ish mean and 90ish STD, same as original post. So can confirm that these stat distributions are accounting for bad cover distributions in your pulls already. 

illusionist_KA

You all making this super complicated.   As a super hoarder. These calculations are very misleading.  

260 pulls is what the minimum was to cover each 5* to 13 covers. However,  the color spread was not ideal and you needed swaps to get the color of choice.   That's where 300 pulls came from.  That became the ideal number without swaps.  So. You need 300 pulls to complete 3x 450.  

You're still relying on RNG, and predetermined pulls.  Either it will go your way or it won't. 

justsing

jsmjsmjsm00 said:

I've run a quick calc that doesn't account for cover distribution and hit a mean of 320ish pulls to cover the three 5s with no other shards or covers. I'm going to assume then, since this post presents a mean higher than that (390), that it is accounting for cover distribution.

EDIT: Ran my own simulation to confirm the case accounting for cover distribution and got 390ish mean and 90ish STD, same as original post. So can confirm that these stat distributions are accounting for bad cover distributions in your pulls already. 

Yes, as confirmed by your simulations, my results account for cover distribution.

Like I mentioned in the Discussion section, the 300 pulls number that gets mentioned a lot came from a time where CS would do 1:1 cover swaps for any Latest 5*, so you wouldn't have to worry about cover distribution. That's gone now, so you'd really want closer to 400 pulls (if you're not getting shards/covers elsewhere) to account for uneven cover distributions.

justsing

illusionist_KA said:

You all making this super complicated.   As a super hoarder. These calculations are very misleading.  

260 pulls is what the minimum was to cover each 5* to 13 covers. However,  the color spread was not ideal and you needed swaps to get the color of choice.   That's where 300 pulls came from.  That became the ideal number without swaps.  So. You need 300 pulls to complete 3x 450.  

You're still relying on RNG, and predetermined pulls.  Either it will go your way or it won't. 

260 pulls will give you, on average, 39 5* covers. Doesn't mean they'll be evenly split among the three 5*s. To get at least 13 covers for each 5* (regardless of color), you'd want closer to 320ish pulls (like jsmjsmjsm00 said). So no... 300 pulls isn't necessarily enough to complete 3x 450...

jsmjsmjsm00

Yup agreed. 

This is definitely not making it super complicated. This is as straightforward as can be. 

Get 13 covers for 3 5*s: 320 pulls
Get champ-able spread for 3 5*s: 390 pulls
Get champ-able spread for 3 5*s starting with two boss event covers and 1000 shards: 250 pulls

Really, the 300 number is just old hat. Pick the new number to hoard to based on the parent post and you should be good to go.

I'm happy to see a post that is just math no conspiracy. 

hilsen2

Great Job on this

Could you adjust the program so you only pull until the the oldest 5* is champed?

Otherwise, you are feeding champ covers to 2 out of 3 characters while chasing the last 5*.

I usually pull about 70 to 100 covers per month to keep my LLs champed before they leave latest.

Lately a few have escaped Un-champed with <250 Shards required

Boss covers are helping reduce this number a lot 

justsing

hilsen2 said:

Great Job on this

Could you adjust the program so you only pull until the the oldest 5* is champed?

Otherwise, you are feeding champ covers to 2 out of 3 characters while chasing the last 5*.

I usually pull about 70 to 100 covers per month to keep my LLs champed before they leave latest.

Lately a few have escaped Un-champed with <250 Shards required

Boss covers are helping reduce this number a lot 

These are the results for just champing one out of the three Latest 5*s:

Scenario	Mean (SD)	Median (P25, P75)	P5, P95
Boss + SCL10 PVE	194 (66)	186 (147, 232)	102, 315
SCL10 PVE only	236 (72)	228 (184, 279)	133, 367
Boss covers only	264 (89)	251 (201, 312)	143, 427
None	307 (95)	295 (240, 360)	176, 480

ETA: I ran these numbers just because I was curious. However, I don't recommend reading too much into them. It's better to follow jsmjsmjsm00's suggestion below.

jsmjsmjsm00

Tldr: if you want to pull each month you need to do 1/3 of the pulls per month that OP lists. 

Simulating exactly what you say may be difficult because of how much that method depends on hysteresis. I don't mean difficult in the sense that we can't simulate it; I mean difficult in that what we get out of it is probably not going to be helpful to what I think you want. 

By this I mean that starting from the cases OP outlines, basically no covers, isn't really the situation you are in. You are basically already at the steady state for these monthly cycles. So running this simulation but stopping at oldest character being champ-able isn't applicable. EDIT: see OP's previous comment. No offense, but I'm assuming that is not useful information for the people that pull monthly. 

Instead, we can consider that the statistics in the parent post are independent of when in the cycle you make pulls; that shouldn't affect the stats. The expectation is therefore that pulling monthly, you'd need to make 1/3rd of the pulls as outlined in the parent post, since there are three months each character is in shop. 

If you are starting this method fresh, you will very very likely miss champing the first two characters, but the third character should follow the same odds over those pulls over three months as outlined in parent post. 

The risk of this method is that coming in below average on a hoard will affect more characters, which could be what you are seeing. For example, say you come in below average after pulling 390 all at once. You will miss champing 3 characters. If you are instead pulling monthly, and say already at steady state, those 390 pulls over 3 months that come in below average would affect 5 characters instead.