## Introduction

Hello Everyone.

I always dealt with the question of how high the drop rate of the cards ingame is. Being one of the older players that played since alpha, I still remember that getting cards was rough and had to do with a lot of luck, since there was no card trader and also no pity points ;)

I collected some data and I'll try now to theoretically calculate the current chances of getting a card. Be aware of that I have definitely no insights into the gamecode and thus can just guess any rates. Also it's worth to mention that I only picked a tiny bit of data to make my statistics about, since everything else would take a lot of time I do not want to invest.

It's more a topic of fun-n-sis and discussion - or maybe some realizings from your side? I am very curious!

The numbers I took from is for the most part from Ranked games and from Higher Elo players (High Platinum and above) - I am very very sure that these numbers will differ a lot depending on ranked or classic mode and also on low to high elo, since in ranked the games go longer and in higher elo the players have tendentially more workers.

But we will see. Right now (having only collected data, but didn't evaluate them yet) I'd assume, that the card drop rate per unit is 0,01%. Why you ask? Well, it's a nice number for something very very rare ;) Back then in Ragnarok Online "cards" had that drop rate too and for me it feels like the same Drop rate in LTD2 hahaha ^^ But we will see on that Later !

Oh, and for the TLDR People or anyone who starts puking on consuming too much maths - Just go ahead and skip to chapter "Summary" ;)

## Collecting Data [Pity Points]

Soo ... where to begin?

I think the probably best beginning point is taking a look at the pity points of each players.

As the ingame prompt says, I quote:

"Each pity point increases the chance of finding a card by 2% (+100% = 2x chance). Every game you finish without finding a card, you gain a pity point. Once you find a card, your pity points reset to 0."

Which means that 50 pity points double the card drop rate (+100%), 150 pity points quadruple the card drop rate (+300%), 350 pity points octuple the card drop rate (+700%) and so on.

Ofc the card drop rate is still very random, right?

So let's start to try to find any similarities.

I for myself took 90 players from 3 large active guilds, which most players priorize on rankeds.

From these players each player had 122,6 Pity points in average.

The highest value I collected was 440, the lowest was 0. Lol ;)

I think it's fair to say, that when you have 122,6 Pity points in average, it takes about the double amount (in average) to finally get that card. So let's assume upon hitting 245,2 Pity points, you finally earn your card. Yay. But what does that in fact mean?

## Collecting Data [Units]

Let's go on.

As we know, the card dropping does not happen after, but inside the game. The event get's triggered after killing a unit.

But for knowing, how many units there potentially are per game "to be killed" we have to take a look at the average game length.

The source https://legionlord.com/statistics tells me, that the games statistically ends between wave 14 and wave 15 in average.

So let's collect the unit kill count, shall we?

First off, let's start with the wave Units:

Wave 01 - 12x Crab

Wave 02 - 12x Wale

Wave 03 - 18x Hopper

Wave 04 - 12x Chicken

Wave 05 - 9x Scorpion (8 normal and 1 miniboss)

Wave 06 - 6x Rocko

Wave 07 - 20x slime (each bigger slime turns into a smaller slime, both can drop cards)

Wave 08 - 12x Kobra

Wave 09 - 12x Carapace

Wave 10 - 1x Granddaddy

Wave 12 - 12x Mantis

Wave 13 - 6x Drill Golem

Wave 14 - 12x Killer Slug

Wave 15 - 4,5 Quadrapus (it's 9 units, 8 normal and 1 miniboss, but halfed, since the game ends 50/50 on 14 & 15)

So in average per game we kill about 148,5 units.

But wait, there's more. It's called Mythium and Sends.

However, I statistically collected the average mythium gain per player on a Wave 14 Ending and on a Wave 15 Ending, also I took the same amount of winners as I did for losers. For this, I took the average of each 20 matches.

For Wave 14-Ending, I got 1351,7 Mythium in Average (976 Lowest, 1759 Highest) and for Wave-15 Ending I got 1630,9 Mythium in Average (1282 Lowest, 2281 Highest).

The average of both Wave 14 & Wave 15 is 1486,3 Ending Mythium

And here is where it get's really hard, since I don't have accurate statistics, so there will be a lot of guessing from my side. I hope you can share my thoughts, if not, make your own calculations with your thoughts :)

I personally assume, that from these mythium values the King get's about half of the available upgrades. Meaning when there are 30 Upgrades available (10 dmg, 10 heal, 10 aoe) I'd say that in average 15 of these 30 upgrades are done, which cost 20 Mythium each.

That means that we have to remove 15x20=300 Mythium from the calculation.

1486,3 - 300 = 1186,3

Here's the next assumption. Sorry for that. But what is the average person sending in average? Many people send many snails for income. Some save and make bigger sends. I personally think that the average send is about 50 Mythium. In early game, a lot of snails are sent; in mid game, a lot of dragon turtles, lizards and dinos are sent and in late game, some high-end units are being sent.

That means, that in average each game about 23,73 units are sent in total.

We have to add these up to our previous:

148,5 "General Units" + 23,73 Sent units = 172,23 Killed units in average, each game

## Let The Calculation Begin

So, as we have calculated, it takes about 245,2 Pity Points / Games to gain a card, so if these numbers are correct then each game you have a chance of about 0,407% for getting a card in average.

Also, it takes you to kill 245,2 (Games) x 172,23 (Units) = 42.230,796 Units in total for each card, meaning that each unit kill you have a chance of 0,00236% to get a card.

Dang that's a lot lower than my first 0,01% prediction :( But still, this is not the initial factor, since it still includes the benefit of pity points. There are some really smart ways to calculate these out and I'm certain that I learned these in school. Sadly, I forgot how to do them :( So I'll do it the long way.

I define the drop rate as "D", with the initial value of "?". As we know.. Each Pity Point "P" is 2% additional droprate, blah blah blah ^^

So let's go. I'll leave some entries out "{ .... }" ;) Else your mousewheel catches fire on scrolling.

Game 001: D * (1 + (0P / 100) = 1,00

Game 002: D * (1 + (2P / 100) = 1,02

Game 003: D * (1 + (4P / 100) = 1,04

Game 004: D * (1 + (6P / 100) = 1,06

Game 005: D * (1 + (8P / 100) = 1,08

{ .... }

Game 050: D * (1 + (98P / 100) = 1,98

Game 051: D * (1 + (100P / 100) = 2,00

Game 052: D * (1 + (102P / 100) = 2,02

{ .... }

Game 240: D * (1 + (478P / 100) = 5,78

Game 241: D * (1 + (480P / 100) = 5,80

Game 242: D * (1 + (482P / 100) = 5,82

Game 243: D * (1 + (484P / 100) = 5,84

Game 244: D * (1 + (486P / 100) = 5,86

Game 245: D * (1 + (488P / 100) = 5,88

I'll leave those 0,2 rest out, since it is already complicated enough. Sigh ^^

Now we have add the "modified" values up, divide them through the number and calculate those against the "initial" "unmodified" value.

So it goes: 1,00 + 1,02 + 1,04 + 1,06 + 1,08 + {....} + 1,98 + 2,00 + 2,02 + {....} + 5,78.. and so on. I think you get the point. I'll use Excel for that.

The sum however is 848,7. If we divide that by the amount (of games) 245, the result is 3,46

This means, that our initial value, after 245 games, was multiplied by 3,46 Times in total.

In order to get to the original number, we have to divide to divide our "initial" Unit-Kill-Card-Dropchance-Number by that number. In calculation, I take the unrounded values in order to not falsify the result more than needed.

Here we go:

Drop Chance per Game: 0,407% / 3,46 = 0,11763%

Drop Chance per Unit: 0,00236% / 3,46 = 0,000682081%

Games needed in average: 848,7

Unitkills needed in average: 146.610

## Summary

So I have no idea if you could follow my calculations.

Also it's worth mentioning that I'm not a superb Brainiac in terms of maths. Probably all of this is just wrong :D Probably not. I'm just an average nerd who likes to play with numbers.

However, here's the summary of my .. "clamings":

- The Rule:"Each pity point increases the chance of finding a card by 2% (+100% = 2x chance). Every game you finish without finding a card, you gain a pity point. Once you find a card, your pity points reset to 0."

- All players have a combined average of 122,6 pity points

- An average game consists of 172,23 killed units for each player

- A card drops in average all 245,2 games in average (That's a 0,407% chance each game)

- Assuming, pity points would not exist, a card would drop all 848,7 games in average (That's a 0,11763% chance each game)

- A card drops in average all 42.231 units killed in average (That's a 0,00236% chance for each unit killed)

- Assuming, pity points would not exist, a card would drop all 146.610 units killed in average (That's a 0,000682081% chance for each unit killed)

So going the opposite direction, you'd had the following card drop chances for your next game, if you had the following pity points:

[Formula: (100/848,7) * (1 + "Bonus Multiplicator - e.g. 0,02 ... 0,04 ... etc")

0 Pity Points: 0,118% Card Drop Chance for this game

50 Pity Points: 0,236% Card Drop Chance for this game

100 Pity Points: 0,353% Card Drop Chance for this game

150 Pity Points: 0,471% Card Drop Chance for this game

200 Pity Points: 0,589% Card Drop Chance for this game

250 Pity Points: 0,707% Card Drop Chance for this game

500 Pity Points: 1,296% Card Drop Chance for this game

750 Pity Points: 1,885% Card Drop Chance for this game

1000 Pity Points: 2,474% Card Drop Chance for this game

1500 Pity Points: 3,653% Card Drop Chance for this game

2000 Pity Points: 4,831% Card Drop Chance for this game

.... You won't reach these anyways, so why continue? :D ...

You can find an entire list here: https://pastebin.com/1EdJWN9J

However.. I had fun calculating these, I hope you had fun reading aswell.

Let me know about your thoughts and give me some feedback.

If you don't agree with these numbers, please let me know your corrected way of mathematics and solution(s). I am very curious!

Source: https://steamcommunity.com/sharedfiles/filedetails/?id=2823687584

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