Examination of heirloom probability of single packs [ Edited ]

[Anjunakrokus]

While listning to a Gaming Merchant pack opening video I coded a little program to estimate the average number of pack that you need to open to get an Heirloom (since you are guaranteed to get it after 499 packs without shards this is per definition less than 500 packs).

Assuming the probability of getting an heirloom from a single pack (which is not the 500th) is constant.

Results:

There are two probabilities at play: the sinlge pack probability p and the final pack probability.

The single pack probability is your chance of opening heirloom sharts if it isn't your 500th pack

The final probability does thake the 500th pack into account.

Thus per definition ON AVERAGE you need ot open less than 500 packs.

if p = 1/100 then you need to open around 100 packs on average, and nobody reaching 500 packs

If p = 1/500, it's 320 packs on average, and roughly 40% reach pack 500

if p = 1/1 000, it's 390 packs, and roughly 60% reach pack 500

if p = 1/2 000, it's 440, with 80% reach 500

if p = 1/5 000, it's 475, with 90%

if p = 1/10 000, it's 490, and 95%

below p = 1/100 000, it's essentially always on the 500th pack

Conclusion:

Considering there's enough fluctuation that you can get heirlooms on your first, 50th, etc pack we can conclude that the real single pack probability lies around 1/500 to 1/2 000. Which is high enough that you could get it before your 500th pack, but low enough that a significant portion will have to open 500 packs. 

The added code is written in python. Change the extension to .py (Heirloom_probability.py) and you should be able to run it. It's kind of slow sadly.

EDIT:

I've found the theoretical curves for a single pack probability p and limit N (Heirloom shards have N=500, Legendaries have N=30).

(The derivations rely on the geometric series and are relatively straightforward).

The average is equal to: average = (1 - (1-p)^N)/p

The fraction is equal to: F = 1 - (1-p)^(N-1)

For Legendaries we know from the Apex Store that p=7.4%, N=30. From which we find that:

average packs to get a legendary: 12.17

fraction of pack before limit (30 packs): 89.24%

[sceptic-noob]

I'm always happy to see a fellow nerd Warms my heart really hehe, I keep spending my time on stuff like this all the time so its nice to know I'm not alone :D

[Axs5626Sxa5001]

@Anjunakrokus

I appreciate this and any “examination” that educates the playerbase. The lack of transparency from Respawn/EA makes things like this of greater value.

[DarthValtrex]

@Anjunakrokus This is not how RNG work though.

The odds are always 1-500 packs or so. The reason being is the total number of outcomes will always be the same. Probability plays no part in RNG's.

Casino games such as slots work off of a similar system. It all boils down to stopping the RNG at the right time, some people never will and some people will do it multiple times. It does not matter if you've opened 10,000 Apex Packs or just 10, the odds are always the same.

[sceptic-noob]

Technicaly yes, but realisticly if you check the statistics its different. I think this is what OP is trying to model here.

[Anjunakrokus]

I am trying to model the probablility of getting heirloom shards when you open a single pack.

Like I said in my post, I'm assuming that as long as you haven't opened 499 packs without getting heirloom shards there is a fixed probablility p of getting heirloom shards. But since you are guaranteed to get them after opening 499 packs without them this has a particular effect on the average number of packs needed.

Mathematically this has to do with the fact that the probability of getting shards from a single pack is not independent from how many packs you've openend.

Now imagine that you get to open 500 packs and your last pack just happened to have had heirloom shards (or you start on a new account).

Worst case scenario: after opening 499 packs without shards, the final pack contains heirloom shards. So the worst case scenario is that you have 1 heirloom pack in 500 packs.

A much better scenario would be that after 150 packs you get heirloom shards. The counter resets and you are no longer guaranteed to get heirloom shards in the remaining 350 packs. But if you are lucky you might get heirloom shards after 250 packs. Thus getting two heirlooms in 500 packs.

So:

Worst case scenario: 1 heirloom in 500 packs

Some lucky baster might just get 2 heirlooms in 500 packs.

Thus ON AVERAGE it is no longer 1 in 500 packs. It must be better than this since the worst case is 1 in 500. So it could be 1 in 499 or 1 in 450 packs (for example).

Importantly:

We have assumed thst there is a fixed probability p of getting heirloom shards if you have NOT opened 499 packs without getting shards.

We enforce that after 499 packs without shards you are guaranteed to get shards on your 500th pack.

Now we, the community, do not know the value of p. We only know that it is less then 1% and that it's larger than 0% (some people get heirloom shards before opening 500 packs).

The plot shows how changing the value of p impacts the average packs you need to open to get shards (if p = 100% you would get heirloom shards every single time, but if p=0% you have to wait 500 packs) and what fraction of players would have to open exactly 500 packs.

From there on we can use the following observations:

1. Many people need to open 500 packs before getting heirloom shards
2. There are multiple people which have gotten heirloom shards before opening 500 packs.

This leads to us putting bounds on the single pack probability p between 1/500 and 1/2 000 (or 1/5000 or so)

This does not mean that equals the number of packs you need to open it just inpacts it.

The average packs that you would need to open to get shards is then still close to 1 in 500.

If the probabilities themselves are not your thing, I hope that youtube tutorials could explain them better then I can.

[pandareno1999]

This is some good work here man.  As I see it from just observation, the chance MUST be far lower than 1/500.  Else we'd see posts around the net all the time from people getting them very early, given the number of players.  Not so many such reports are received, only enough to know that it is possible.

I recently received heirloom shards.  If it wasn't my 500th pack, it was very close.  I knew for certain that I was within ~30 packs of 500 when I got it (not that my own anecdote is terribly relevant).

[hayhor]

I've gotten shards twice in less than 150 packs. 😀

[OldTreeCreeper]

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@Anjunakrokus wrote:

.

Mathematically this has to do with the fact that the probability of getting shards from a single pack is not independent from how many packs you've openend.

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@Anjunakrokus my brain hurts  I read and reread your posts, and even googled a bit to see if I could grasp what you were positing. And no it's clearly beyond my pay grade.

How can it not be Independant, as a packs number opened is a number lost into the ether and has no bearing on the next packs outcome? Other than pack 500. And with packs are we not talking about whole numbers? 

To add to your probability sums, I got my shards for €43 as part of the anniversary event last year, combined with lots of crafting materials and a few spare coins and a ea play  membership (€1) giving a shop discount of 10%.

[sceptic-noob]

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@OldTreeCreeper wrote:

@Anjunakrokus my brain hurts  I read and reread your posts, and even googled a bit to see if I could grasp what you were positing. And no it's clearly beyond my pay grade.

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Thats the beauty of it, but the results can be read, there is a feact that there is a fixed chance of getting shards from 1-499 apex pack opening, but the fact that you have a 100% chance on 500 makes that a thing. The chance technically dont change if you are on pack 1 or pack 499 but you can add dimentions to this and calculate a statistical spread on where most people get their packs and create statistics on this. So @Anjunakrokus tried to calculate his version of this and explained what he did to do this, if you are a statistician you can evaluate or even suggest a different approach or if you are not you can evaluate and read the results and just enjoy that someone actually took the time to do this, I think its awesome :D