The Vault! Some basic statistics

halirin

There seems to be some confusion about the vault and the statistics thereof. Since I hate working on my dissertation, I thought I'd take this opportunity to talk about the statistics in a video game on the internet!

https://www.dropbox.com/s/ut58a2cd2drt4 ... .xlsx?dl=0

This is a link to an excel file which shows the probability of drawing at least one 4* character (or item) out of the vault. (4th column)
It also shows the probability of drawing at least one 4* prize under the normal system (if we assume the same 5/300 odds) (5th column).
Finally, I calculate the difference between the two cumulative odds. It's largest when you're cashing in 111 tokens (5.7% better odds for having used the vault instead of the old system). That's because the odds of getting zero 4* prizes just get vanishingly small under either system once you're using enough tokens.

But if all you care about is 4* prizes, then the idea of hoarding 300 tokens to cash them all out at once is very likely sub-optimal. Once you get the first 4* prize in a week, you should probably stop using tokens, because you've just dropped your probability of getting a 4* on subsequent draws by ~20% (from 5/N to 4/N-1). If that 4/n-1 is less than 5/300, then you should just wait for the next week. And that's an important consideration because you're probably gonna get some 4* prizes sooner than you think! At 40 tokens, your chance of getting at least one 4* via the vault is 51%.

If that last number seems high to you, it's because you haven't heard of the birthday problem. Lucky you! I wish I got to learn about this for the first time today, but that opportunity is all yours. Basically: when you have N people, the chance that at least two of them share a birthday is 100% when N is 366 or above (so far so intuitive). But when N = 70, the probability of a birthday match is 99.9%. In fact, with just 23 people, you have a 50% chance of a match. Wikipedia lays it all out: https://en.wikipedia.org/wiki/Birthday_problem

Of course, things get a little trickier if you're also interested in the 3* prizes. If there's any interest I could write about the probabilities for getting at least X number of Y desirable items and optimal strategy there. But it's likely that the best answer will be to get to at least 30 to 40 tokens, and just wait until a vault in which you want "a lot" of the 3* and 4* prizes.

So none of this is to say that the vault is a great/terrible system or that the devs are/aren't evil and using chemtrails to cause AI cascades that make you buy health packs. But with any luck, maybe we can use probability theory to help each other make good hoarding decisions (protip: skip tokens; hoard gas and groceries if you're in Greece).


Edit: People (correctly) pointed out that the 4* rewards that aren't covers really aren't that big of a deal, so here's the same spreadsheet with only 3/300 target items (the three 4* covers) and slightly clearer column headings:

https://www.dropbox.com/s/1qp0kiqpfvt9u ... .xlsx?dl=0

You can see that the max difference between the vault and non-vault occurs later (pull 174 tokens in the vault). But also, the vault's advantage just gets bigger (compared to the case using 5 targeted items) after about 70 tokens, and gets to be almost twice as big (10 percent!). What this means practically, is that if you hoard a whole bunch of vault tokens, you're decently less likely to have a 4* drought with the vault system. We knew that was true at 298 tokens (0% chance of **** drought in vault, 5% chance in normal 1/100 tokens), but it's also noticeably true with "only" 50+ tokens.

Double edit: Yeah, the graph grumpysmurf posted!

Unknown

Amazing spreadsheet! People do mind though it is build on wanting 4* rewards, not heroes. I doubt anyone cares for 10k Iso, and 1k HP for some people is just 40% of 4* cover which may or may not be 'valuable enough'. Few observations from me:

One, the example from Birthday Problem doesn't correlate with drawing mechanics as you're not looking for any two draws being same but for one of draws to be same as you predetermined (difference between searching for at least two people sharing date vs searching for at least one person matching birthday of five you already chosen beforehand). And I believe you can only apply this logic for determining 'what would be without vault', seems it is used that way in your spreadsheet.

Two, while hoarding 300 tokens and opening them at once isn't most effective strategy (all you're doing is flattening RNG to zero), there is no point of cashing them early. For example, opening 40 tokens gives you ~50% chance of getting one 4* reward, but also not getting any, a flip of a coin. Lets say we're unlucky and open three vaults with bursts of 40 tokens and receive no 4*. The chance of that happening is 12.5%, fairly high. Meanwhile, if you'd save 120 tokens and use them on single vault, the chance of not getting one 4* is merely ~7.5%

And finally, optimizing vault efficiency is really easy and don't really require much math. For me, the only valuable things in the vault are the three **** heroes (I count 1000 HP more of a bonus, and 10k Iso is just 40 Moonstones). 300/3 = 100. What I'm expecting is getting one cover in first 100 draws, second in another 100, and third in last 100. First thing is to hoard 300 tokens, then start opening one by one until you draw a 4*. If you used less than 100 tokens to that time, stop and let vault reset, you are on good side of RNG. If you haven't, continue drawing until second 4*. Did you spend less than 200 tokens? Stop. Not? Unlucky, continue opening till third 4*.

This way you optimize vault mechanics to maximally increase your chance of covers and at absolute worst (last cover on 300th token) come to an 'expected' 1% draw chance.

Raffoon

Great table Thanks for laying down some math.

It seems when you count 4* prizes, that you're counting the 10k ISO and 1k HP. I suppose that's one valid way to look at it, but personally I'd really only be concerned with 4* covers. I'd be interested to see how this shakes out with 3 items of interest instead of 5. (Though it's probably fairly similar in most regards)

I'd comment further, but have to run. Looking forward to examining more later today.

GrumpySmurf1002

I didn't put the full table in, but the graph here is same math as OP, with 3 rewards instead of 5.. If I get bored later I'll update it with the secondary line.

viewtopic.php?f=7&t=30889&start=140#p376648

Malcrof

Amazing sheet, but here is the rub.. my very first token was a 4*.. so how bad are my odds of getting either of the other 2 4*'s.. which is the only reason i am still opening and not hoarding..

GrumpySmurf1002

Malcrof wrote:

Amazing sheet, but here is the rub.. my very first token was a 4*.. so how bad are my odds of getting either of the other 2 4*'s.. which is the only reason i am still opening and not hoarding..

Pulling 1 on the first token, it's 5.939% to pull another one in the next 9.

evil panda

In before someone inevitably tries disproving mathematics with....histrionics

simonsez

Malcrof wrote:

Amazing sheet, but here is the rub.. my very first token was a 4*.. so how bad are my odds of getting either of the other 2 4*'s..

Not sure, but the way luck balances out, I'd say your chances of getting hit by a bus are excellent!

GrumpySmurf1002

Updated using Vault odds, 1% pull non-Vault. Benefit maximized at 174 tokens.

mikepro

so far 3 pulls for me and best i got was 2 3* which i needed which is what i wanted. now the 1000HP would help me out a lot cause i've been playing for 90 days as of today and the extra coins would help for more team members since i already have about 900 saved so that would be 4 more slots open or closer to a 40 pack with higher rates later on. but i'd love it for the slots so i can have the required character so i can earn 2 tokens as opposed to 1 token a day.

DaveR4470

Math!

I think there might be some mixing of apples and oranges here. Correct me if I'm wrong...

For non-vault pulls the chance of pulling a 4* is fixed per pull. Therefore, your column for non-vault pulls is, basically, the probability that after X number of picks, you will have picked at least one 4*. I'm with you there.

Your vault column reflects a variable probability due to a shrinking pool. So each column tells you the probability of pulling a 4* with token X if you have not drawn a 4* with the previous (X-1) tokens. I'm also with you there.

However... those are not the same thing, and not really comparable.

The problem is that assuming you have not picked a 4* in the previous (X-1) tokens is assuming a probabilistic "worst case". In other words, you're comparing an increasingly statistically unlikely case (an X-1 run in the vault without a 4*) with a statistically fixed case (the pre-vault fixed percentage stats).

The proper comparison, I believe, is the odds that X tokens from the vault will yield AT LEAST one 4* vs. the odds that drawing X tokens pre-vault would yield a 4*. So the proper comparison for, say, 30 tokens is the probability of drawing a 30-pull set within all possible 30-token draws from the vault that contains at least one 4* vs. 39.6019%, which is the expected 4* pull rate for 30 tokens pre-vault.

Bottom line, I think you might be understating the statistical advantage of the vault.

DaveR4470

Also -- and I'm not sure how you'd quantify this, exactly -- there's a DISadvantage to the vault that's not being mathematically expressed. Namely, if you have 30 regular tokens, there is a statistically non-zero chance that you'd pull, say, 17 4*s. For the vault, that's completely impossible.

So for X>3, there is an inherent statistical disadvantage to the vault that increases as X approaches 300. At 300 tokens exactly, the disadvantage would be the difference between 3 (the total 4*s you'd get from the vault) and the statistical mean expected number of 4*s received for X=300. Which is reasonably calculable, but I'm too lazy to do it.

So! Much! Math!!!!!

GrumpySmurf1002

DaveR4470 wrote:

The proper comparison, I believe, is the odds that X tokens from the vault will yield AT LEAST one 4* vs. the odds that drawing X tokens pre-vault would yield a 4*. So the proper comparison for, say, 30 tokens is the probability of drawing a 30-pull set within all possible 30-token draws from the vault that contains at least one 4* vs. 39.6019%, which is the expected 4* pull rate for 30 tokens pre-vault.

Bottom line, I think you might be understating the statistical advantage of the vault.

p(>=1) = 1-p(0), which is what is being calculated. (check the formula in his columns). Nothing wrong there

Your point on p(>3) is a fair one though. On 298 pulls, you're about 18% to have a 4+ yield from a traditional token, whereas that obviously doesn't exist with the vault.

And yes, the math is more complicated, and I don't feel like digging out my stats books to refresh my memory on how to merge the two ideas.

DaveR4470

Ah, okay, I see it. The formulae confused me a bit, but I think it's correct.

Malcrof

we need a sub-forum for Math, Statistics and Probabilities.

Pylgrim

DaveR4470 wrote:

Also -- and I'm not sure how you'd quantify this, exactly -- there's a DISadvantage to the vault that's not being mathematically expressed. Namely, if you have 30 regular tokens, there is a statistically non-zero chance that you'd pull, say, 17 4*s. For the vault, that's completely impossible.

On the other hand, having complete, visible information on the quantity and nature of prizes, allows you to stop when you got all you want. Under the old ways, if you got three 4*s at the beginning, you may feel tempted to keep opening tokens just to see if your luck keeps up. With the vault, you know that you must stop and save your tokens for the next vault.

AXP_isme

Malcrof wrote:

we need a sub-forum for Math, Statistics and Probabilities.

Judging from this thread it would be one of the more popular.

GrumpySmurf1002

Malcrof wrote:

we need a sub-forum for Math, Statistics and Probabilities.

I'll miss you guys.

DFiPL

augh math no god why i majored in history i was told there would be no math

Malcrof

DFiPL wrote:

augh math no god why i majored in history i was told there would be no math

History is mostly about remembering numbers (dates, etc...) ....

CaptainFreaky

Malcrof wrote:

we need a sub-forum for Math, Statistics and Probabilities.

This is immediately what I thought as I read through this...

We could easily have a sub-forum on Math, Statistics and Probabilities jokes!