I previously ran a super quick simulation on the number of Legendary Token (LT) pulls needed to cover a specific Latest 4*: Update to 4-Star Pack Odds (6/4/18) - #120 by justsing - MPQ General Discussion - 505 Go! Official Forums
Now I’ve run a more complex simulation on how this change in odds (removing the higher odds on the 12 Latest) has affected the ability to cover a 4* through LT pulls.
There are two parts to this analysis:
- The number of pulls needed to cover any 4* vs. a specific Latest or Vintage 4*
- The number of covered 4*s given a set number of pulls (300 and 700)
All the simulations were conducted with 10000 runs. I assumed that all characters were starting with zero covers, and there was only one 4* BH character at a time.
I know 4s can also be obtained through other means, such as Heroic tokens, PVE, PVP, etc. This analysis does not account for that, so the actual rate of covering 4s should be better than the results I present here. However, I do think it’s useful to isolate and see what this change has done to 4* acquisition via LT pulls specifically.
Some terminology before I start:
- old = older odds where Latest had higher odds than Vintage 4*s
- new = current odds where all 4*s have the same odds
- latest BH = a single Latest 4* has been selected as the BH
- vintage BH = a single Vintage 4* has been selected as the BH
The code for this simulation can be found here: MPQ 4 star pulling v2 - Pastebin.com
PART 1
The goal of Part 1 was to see how the change in odds affected the number of pulls needed to cover any 4*, a specific Latest 4*, and a specific Vintage 4*. Here are the distributions of the pulls:
The tables are under the cut for anyone who wants to see the actual numbers:
To cover any 4*:
4* Odds in LT
|
Mean Pulls Needed (SD)
|
Median Pulls Needed (P25, P75)
|
Old
|
250 (44)
|
250 (220, 280)
|
New
|
523 (79)
|
528 (472, 578)
|
Old + Latest BH
|
186 (46)
|
184 (153, 217)
|
New + Latest BH
|
275 (82)
|
266 (215, 325)
|
Old + Vintage BH
|
233 (44)
|
233 (203, 263)
|
New + Vintage BH
|
275 (82)
|
266 (217, 325)
|
To cover a specific Latest 4*
4* Odds in LT
|
Mean Pulls Needed (SD)
|
Median Pulls Needed (P25, P75)
|
Old
|
433 (135)
|
415 (337, 511)
|
New
|
1176 (369)
|
1126 (914, 1389)
|
Old + Latest BH
|
198 (62)
|
189 (153, 233)
|
New + Latest BH
|
277 (87)
|
267 (216, 325)
|
To cover a specific Vintage 4*:
4* Odds in LT
|
Mean Pulls Needed (SD)
|
Median Pulls Needed (P25, P75)
|
Old
|
1923 (604)
|
1842 (1497, 2261)
|
New
|
1180 (371)
|
1128 (917, 1387)
|
Old + Vintage BH
|
306 (95)
|
294 (238, 359)
|
New + Vintage BH
|
275 (87)
|
263 (214, 323)
|
My main takeaways from this:
- BHs (when used one 4* at a time) greatly reduce the number of pulls needed to cover that specific 4*, so choose your BHs wisely!
- Aside from covering specific vintage 4*s, the older odds generally required fewer pulls to cover a character as compared to the new odds
- Among these options, the fastest way to cover a 4* was to BH a Latest 4* during the older odds period
- Now, under the new odds, you need about 523 pulls (sd = 79) to cover any 4*, 1176 pulls (sd = 369) to cover a specific 4*, and 275 pulls (sd = 82) if you BH that 4*.
PART 2
The goal of Part 2 was to see how the change in odds affected the number of 4s you could potentially cover in a set amount of pulls. I used 300 pulls since that’s about how many Latest pulls you need to cover all three Latest 5s. Then, I also looked at 600 pulls as an additional comparison point. For this analysis, I don’t take Latest 4* rotation into consideration, so the numbers here would reflect someone who pulled their hoard all at once as opposed waiting for covered Latest 4*s to rotate out before pulling more.
Main Takeaways:
- The older odds led to more covered 4s in the same number of pulls, which means that the 3-4 transition could probably occur faster.
- In the long term (600 pulls), the newer odds allow for a greater number of 4* characters getting close to 13 covers while the older odds mostly focused on finishing the Latest 12.
- One thing to note, these results doesn’t account for Latest 4* rotation. If rotation was included in the simulation, I’d expect the number of 4s with 10 or more covers to be higher than the current numbers for the older odds under 600 pulls.
All of this suggests to me that 4 progress is back to being a marathon where you’ll very slowly cover multiple 4s in a long time instead of more quickly covering the Latest ones. Whether that’s a good thing or not depends on the state of 4s in your roster and which kind of progress you prefer.
(I know I’ll be grateful for vaulting and for the old odds being around when I was making the 3-4* transition…)
Finally, choose your BHs wisely 
Very insightful. I think it broadly supports what intuition, and the community position, would suggest. Still it’s good to see this in numbers.
What is particularly telling is the conclusion that 300 pulls is enough to fully cover three latest 5s but not to fully cover even one 4 under the new odds. It seems unlikely that anyone would have 300 pulls and not have any 4* covers as a starting position but it’s still a strong statement where dilution is concerned.
I know people complained when it looked like it was more efficient for newer players to skip the 3* tier and move to being 4* players by covering those 4s in latest. On the one hand that won’t be a viable strategy any more but on the other hand it may now be a case of building a 2 roster and then grinding out CP as you get dozens of 1-2 cover 3 and 4* characters until you have enough pulls saved up to jump straight to 5* land.
As a silver lining this may alleviate the iso crunch in the early part of the 4* transition. When you have lots of low covered characters the cost of keeping them levelled should also be pretty low. Best not think about what happens at the business end though.
Back when roster-based scaling was still a thing for PVE, I was initially more in favor of slowly building out my 4s instead of jumping so quickly into 4 land with vaulting, but it turned out that 4* land is super fun. I feel for the newer players who now have to slowly build out their 4* roster. In the long term, they’ll cover 4*s more evenly and have better access to Vintage characters, but it’ll take a long while for them to get there. It’s not a fun wait.
When my other computer is working, I’ll add some numbers on the number of usable covers you should expect to get for a specific 4* given 300 or 600 pulls.
If people are interested/curious, I can also run this same set of simulations with all characters starting at 1/1/1 or some other build. 1/1/1 seems reasonable due to boss events being run more regularly.
The more I think about it the more I realize I started playing this game at the perfect time. Vaulting hit right as had about half the 3s champed and was ready to move on to 4s and because of vaulting I was able to rapidly cover some key 4s and quickly move into the tier. The past few months without vaulting have allowed me to champ just about every character that was in latest during that time and even as I lament the odds now I have very solid 4* roster and am looking forward to what appears to be more regular 4* PVPs.
I feel the same. I had completed the 3 star and was about 12 deep with 4 star champs. Vaulting focused greatly on building new fours to further make my 4 
roster way deeper than before. I also benefitted from unvaulting with the better odds for new releases cause I was able to finish off the previously vaulted characters while also building latest at a reasonable pace.
Now I’ve only pulled 15-20 LTs since the change, but not a single one has been a previous latest and though I have 95% of fours champed it is concerning that we’re back to the dreaded pre-vaulting days of newest releases taking months to cover again.
