How many packs should you buy to limit your chance of bad luck?

Hey everyone! I was recently watching a video by @infidel1258 (

where he says that if you open a few packs, they are essentially lottery packs, but if you open a LOT of packs, the RNG evens out and you will get a well-defined distribution of cards. This is true, and my aim here is to answer what a good definition of a LOT is?

Simulating packs

To answer this, it isn't enough to calculate the expected number, as there is variance around expectations, particularly when the probability is low (think gold foil legendary chances). So to do this, I simulated a varying number of packs, where each card has a 20% chance of being rare, 4% chance of being epic, 1.6% chance of being legendary (assuming leg potions), and then the remainder common. Each card has a 4% chance of being a gold foil (also assuming gold potions).

10 packs

If you buy 10 packs, your chance of getting at least one legendary is ~55%. So with only 10 packs, it's not much better than 50/50 to even get one regular foil legendary.

Below is a single, representative draw from 10 packs. No GF legendaries, epics, or rares, but a GF common. This pull did get a single legendary as well.

100 packs

If you buy 100 packs, you chance of at least one legendary is >99%. So if you buy this many packs, you're almost certainly getting a legendary. So 100 packs can smooth out the variance on the regular foil cards.

But what about gold foil? Your chance of at least one GF legendary in 100 packs is ~28%. So if you're counting on a GF legendary, you need more packs.

Below is a single, representative draw from the 100 packs. Now there's a bunch of GF commons, a fair few GF rares, and a few GF epics, but no GF legendries.

1000 packs

If you buy 1000 packs, you're almost certainly getting a fair few legendaries (expecting about 80 BCX). GF Legendaries? Your chance of getting at least one with 1000 packs is ~96%. So a really good chance of getting at least one GF legendary. On average, you'll get 3.2, but again, because of the very low chance, the variance is high.

Here's the same graph for 1000 packs, and we now see one of every type of card. The bars for the more likely types (all regular foils, and the more common GFs) have nicely smoothed out and hardly changed from the 100 pack graph. But now the GF epics is a noticeable chunk, and we've got a GF legendary or two.

So where's the sweet spot?

TL;DR: 1,000 packs

Based on the above, if you want to treat your packs like they hold a dependable value and not as a lottery ticket, you need to plan on opening at least 1,000. So much of the value is tied into the GF legendaries, and if you can't be sure of getting at least one of those, it's not a good bet.

Below is a graph of the chance of getting NO GF legendaries vs. the number of packs you open. As you can see, it drops quickly until just about 1,000 packs, then levels off as you're then opening enough packs to be confident of getting at least one.

Conclusions

I know opening packs is fun. But if you are opening fewer than 1,000 packs, don't be disappointed if you don't get a GF legendary. As with most TCGs, the best value plan is to buy singles. In a game like this especially, where the supply is limited, even if you buy meta singles at the beginning, they will go up in value over time. So buy and open packs, sure, but treat them like lottery tickets unless you're opening up a lot of them.