What are the chances of you getting an heirloom in Apex Legends?

This is from three years ago, but reading this forum as a maths student just makes me want to reply to this.
RockDokRock​ and vmadman​ are correct, its basic probability - it's not the fact that the 499th apex pack has a 99% probability that it will have an heirloom, it's that by the time you've opened 499 apex packs, per se, there is a 99% probability that you'll have gotten an apex pack by then.

It's the exact same as a coin flip - except lets model it as a biased coin. Probabilities sum to 100% (or 1 if you're dealing with numbers) - so the probability of getting heirloom shards is, 0.045%, and therefore the probability of not getting heirloom shards is 100% - 0.045%, which is 99.955%, rather than a usual coin being 50% and 50%.

Let's open one apex pack - the outcomes are either H (heirloom shards) or N (No heirloom shards)

Outcomes:
H - 0.045%
N - 99.955%

Let's open two now.
Outcomes:
NN - 99.955% x 99.955% = roughly 99.91%
NH - 99.955 x 0.045% = roughly 0.04%
HN - Roughly 0.04%
HH - 0.00002025% (Two heirloom shards in a row - very rare!)

Now, as we can see the probability of getting at least 1 heirloom shard pack is based on the outcomes. there are three outcomes from two apex packs, and to find the probability of getting at least 1 heirloom, we can add the probabilities of the outcomes that have heirloom shards in them as a result - which would be NH + HN + HH - adding then up, we get roughly 0.0802025% - very low, obviously. This is the probability that we will have received heirloom shards after opening two apex packs - not the 'accumulative chance' of getting an apex pack on our second pack.

Let's scale this all the way up to 499 apex packs - I can't show all the outcomes as there are tons of them - 2^499 to be exact! (Which would be very large, and I'm not about to go here writing NNNNNNNNNNNNNNNNNNNNNNNNNNNNN..... HNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN.... and so on.) Using a calculator, the probability of AT LEAST getting 1 heirloom pack BY NOW, after opening our 499th apex pack, would have been... 20.116%! This doesn't mean that we have a roughly 20% chance of getting an heirloom on our 499th apex pack, it means that BY NOW, having opened all these packs, there will have been a 20% probability of us at least getting at least 1 apex pack with heirlooms in it.

Even chatgpt backs me up:

Hope this helps!

P.S. being in high school or having a PHD has nothing to do with this, I'm still in high school (the equivalent in the UK at least - year 12 or 11th grade I think), it's just a matter of understanding the concept of probability.