How Long Would TCG Take to Complete ~ CRAZY MATH - Ramble 25

[00:00:00]Greetings gamers. In this episode I have something that is extremely nerdy for you guys and I actually decided to nerd out and do some math. As expected, that's basically what I do in my free time cuz I have no life.

[00:00:12]And uh what we did is we wanted to math out the trading card game that is on RuneLite. And uh if you haven't downloaded this game, I made a video on it a few weeks ago and it's been actually booming in popularity and I've seen a lot of people do some card lock series. Like I saw Fo started one. I think it's pretty cool, good side series to do. And uh I kind of want to math out some of the the statistics behind this game because it's actually pretty crazy.

[00:00:35]So I am in the Discord and I did reach out to the owner and I asked if they can share drop rates. Unfortunately, they did not share it. They said it was too hard. And also that with the new revision of the beta, they should change drop rates entirely.

[00:00:47]So I kind of wanted to go with the alpha version as we have today and calculate the expected amount of packs you need to open to complete the entire collection.

[00:00:56]And funny enough, this is actually pretty similar to the collection log.

[00:00:59]The only difference of course is uh you're basically hunting the cards instead of collection log slots. And uh what I did is I simplified the math intensely and I actually essentially ignored all the cards asides for the godly.

[00:01:11]So the godly, if you're not aware, are the cards in yellow when you open the log.

[00:01:16]And these cards are the rarest ones you can get. And there's a total of 109 godly cards on the collection you can get. So what I did is I did a basic coupon solver problem and I essentially calculate how many rolls do you need in order to complete all 109 godly. Now these do assume assume that they're at the same exact uh statistical odds each one. At which I believe it is a I read through the cord on the co- cord. I read through the code on GitHub and I was able to confirm, I mean, based on my knowledge and I'm not a professional coder, so uh hopefully I don't I don't screw this up. But there's very likely that my math is wrong here. Uh but I did the best I could. That's why it's called napkin math. And um yeah, so basically by calculating this, I was able to assume uh so for example, this is a coupon solver problem, so the first godly card you get is guaranteed.

[00:02:05]There's like no chance you're going to dupe. The second one you get, there's a chance you can roll a dupe from the previous one, and so on. And then for example, the 10th unique card you get, you have to expect 10.43 rolls. Uh so in my case, I opened mine, and if I look, I have 36 godlies. So if I scroll down to 36 godlies, I should expect around 43 total godly items, so a couple dupes.

[00:02:32]And I actually did the math for myself, and it actually lined up pretty well. Uh so it seems like my math is right according to myself. It would mean I'm slightly unlucky on my side, which I guess it's nothing crazy.

[00:02:44]But if you scroll all the way down to complete the collection of to get all 109 godlies, you need 574 godly card rolls. So that means you get around 400-ish dupes along the way.

[00:02:57]So this is the first section of it. Uh this is actually the easy part, so basic coupon solver. But now if we go into nerdy stuff, there's actually two different packs you can get. Uh you can open the normal pack, which is the basic one, and then you have a chance, which is a 1 in 3,000 per pack, which is extremely rare, to get an apex pack. And this apex pack has a higher odds. So for example, the regular pack, every single card has a 1% chance to get a foil. And I'll talk about the foil later. They're basically like an advanced version or a rare version of this specific card.

[00:03:30]And then the apex card, a 1 in 3,000 per pack, and they have a higher chance of getting you the stronger tiers.

[00:03:37]So if I go to normal completion, uh basically, I calculated for normal pack, the average expected number of the uh godly cards you can get per pack is.033.

[00:03:51]And if you get a godly pack, of course, the odds are much higher. The odds go up to 9.9% chance. So, the expected number of godly cards you get is basically almost 0.5.

[00:04:02]Now, if you do a simple statistical analysis here, you can see that on average, whenever you open a pack, you expect uh godly cards. Now, you can notice that the godly pack did not really have a big impact here because pack is so rare, but you'll see in the foil completion how it has a bigger uh effect.

[00:04:21]So, the total packs required to complete all the godly cards, I calculated around 30. So, that means sorry, I'm cooked. I calculated 30.16. So, if you get 0.033 per pack, on average, you have to open 30.16 packs to get one godly card, which means the expected number of packs you need to open to complete the base collection of the T uh CG game is 17,336 packs. And we can calculate that quickly just by doing times 2,500.

[00:04:55]And that is a lot of zeros. That is 43 million credits you need to complete the base collection. Which is kind of cool, kind of nuts. And it's actually nothing in comparison to the foil. So, I mentioned earlier that when you open a normal pack, there's a 1% chance your card could become a foil. So, that makes the odds incredibly rare.

[00:05:14]And it's a lot harder to calculate for the Apex pack, but I was able to calculate for the normal pack that the odds of getting a foil godly card is around this many cards per pack on average.

[00:05:25]Now, if you want to calculate the Apex pack, this one is a lot harder. I tried to put comments here to explain. I did a lot of this on a piece of paper on the side, and my writing is pretty bad. I don't want to share that. But, there is a 5% chance to get a foil per card, and every card has a 9.91% chance to be a godly in the Apex pack.

[00:05:44]And obviously, you need a one in 3,000 chance to roll the Apex pack.

[00:05:48]And then the gar- there's a guaranteed at least one foil, and this complicates the math a lot. Uh because on average you're expected not really get any foils here since it's 5% per foil. You have a 77.36% chance of not getting a foil, which means you'll get at least a guaranteed one extra foil. So, we can actually mathematically combine all this together, and the expected foil per card that you open is a 20.48%.

[00:06:13]And that means if you combine everything together, uh wait, I got to double-check my math.

[00:06:18]Give me a second.

[00:06:20]Okay, actually never mind. The error was kind of tiny. I it didn't It made a difference, but it wasn't that crazy, I guess.

[00:06:27]Uh so, the actual rate here to get a an expected foil, I'm so cooked. This is so much talking at once. Holy I should have read a script for this. Uh but the expected number of foil uh for an Apex pack is 0.1.

[00:06:41]So, if you combine the stats the odds of each together, you can see that on average you expect to get uh around What is it? How many zeros is that? Like Okay, it's basically you have to open 2,750 packs to get one godly foil card on average. And then as I showed earlier, if you calculate the expected from the coupon solver problem, you need 1.5 million packs in order to complete the foil collection log section.

[00:07:09]And then if we multiply this by 2,500 points, that is 3.95 billion packs or points or I guess credits you need to grind in order to complete the foil collection, which is kind of crazy and pretty nuts. Like that like as far as I know, the best way to get points right now is barraging like slayer tasks and stuff. And you get a little bit over 200,000 points per hour.

[00:07:34]I think you could get more points by creating fresh accounts and just rushing the early game skilling cuz you get so much points for levels.

[00:07:40]And without trading, of course, this is insane. With trading, you can obviously complete this a lot easier just by offering like the trade cards or with GP to buy the ones that you're missing.

[00:07:50]Not sure if that's allowed or not. We're not recommend playing with that. But that's the math behind this entire talk, I guess. So, I hope you guys liked it.

[00:07:59]It's kind of cringe. I for sure made some mistakes here. I did a lot of assumptions and some like um you know, so to ease the calculations, some assumptions there. Uh but overall, pretty cool napkin math. Figured I'd share it with you guys cuz it's kind of cool. And I like yapping. I like doing math. So, this was the perfect cup of tea for me. So, if you enjoyed this yapping, don't forget to like, comment, subscribe. And if someone ever talks about this game, you can tell them, "Hey, have you opened 17,000 packs?" Cuz that's the expected completion. And there was actually one guy in the log chaser's kind who completed the pack.

[00:08:31]Unfortunately, he did a bug abuse. Uh but he it took him around 15,000 packs to complete. So, my math is probably pretty accurate for the normal completion. Uh the four completion, I'm not entirely certain. I would not bet money on this being right or or wrong.

[00:08:45]Uh I would like to someone to maybe double-check my math on their own side.

[00:08:48]And uh yeah. On that note, peace out.

[00:08:50]Hope you enjoy your weekend. Don't forget to like, comment, subscribe. This is all about math. Peace out. Take care.