Examination of heirloom probability of single packs
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%
