Examination of heirloom probability of single packs

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.