## 【HEX】Number of Copies

Who likes statistics?  Another basic HEX strategy article, the HEX 101 series.  In this article, the very basic statistical analysis is provided to indicate the reason of why you want to include 4 copies, 3 copies or perhaps even 2 or 1 copies of specific card.

## 4 copy limit

This is a pretty much standard in a TCG where minimal deck size is 60 cards.  Basically, you can include up to 4 copies of the same card in your deck.

## Why do I want to include 4 copies rather than just 1 copy of 60 different cards?

This is where things become a little bit interesting.  When you have never played TCG before, it seems like the best to have more card types so you can do many more things.  Conceptually this thinking is true, but there is a major flaw with this approach.  The flaw is unreliability.  If you have only 1 copy of a card in 60 cards deck, you don’t know when you will draw the card; hence, your actual deck play will not be able to rely on the card.

## Probability theory

So the concept is very simple.  If you have more copies in your deck, the chance of you drawing at least one of the copy is higher.  Simple, isn’t it?  So just include 4 copies of all card?

 Turn/# of copies 1 2 3 4 1 11.67% 22.15% 31.54% 39.95% 2 13.33% 25.08% 35.42% 44.48% 3 15.00% 27.97% 39.14% 48.75% 4 16.67% 30.79% 42.72% 52.77% 5 18.33% 33.56% 46.16% 56.55% 6 20.00% 36.27% 49.46% 60.10% 7 21.67% 38.93% 52.62% 63.42% 8 23.33% 41.53% 55.64% 66.54% 9 25.00% 44.07% 58.53% 69.45% 10 26.67% 46.55% 61.30% 72.16% 11 28.33% 48.98% 63.94% 74.69% 12 30.00% 51.36% 66.45% 77.05% 13 31.67% 53.67% 68.85% 79.23% 14 33.33% 55.93% 71.13% 81.26% 15 35.00% 58.14% 73.29% 83.13% 16 36.67% 60.28% 75.35% 84.86% 17 38.33% 62.37% 77.29% 86.46% 18 40.00% 64.41% 79.14% 87.92% 19 41.67% 66.38% 80.87% 89.26% 20 43.33% 68.31% 82.51% 90.49% 21 45.00% 70.17% 84.06% 91.61% 22 46.67% 71.98% 85.51% 92.63% 23 48.33% 73.73% 86.86% 93.55% 24 50.00% 75.42% 88.14% 94.38% 25 51.67% 77.06% 89.32% 95.13% 26 53.33% 78.64% 90.43% 95.80% 27 55.00% 80.17% 91.45% 96.40% 28 56.67% 81.64% 92.40% 96.93% 29 58.33% 83.05% 93.28% 97.41% 30 60.00% 84.41% 94.09% 97.82% 31 61.67% 85.71% 94.82% 98.18% 32 63.33% 86.95% 95.50% 98.50% 33 65.00% 88.14% 96.11% 98.77% 34 66.67% 89.27% 96.67% 99.01% 35 68.33% 90.34% 97.17% 99.21% 36 70.00% 91.36% 97.62% 99.37% 37 71.67% 92.32% 98.01% 99.51% 38 73.33% 93.22% 98.36% 99.63% 39 75.00% 94.07% 98.67% 99.72% 40 76.67% 94.86% 98.94% 99.79% 41 78.33% 95.59% 99.16% 99.85% 42 80.00% 96.27% 99.36% 99.90% 43 81.67% 96.89% 99.52% 99.93%

Probability of minimal at least 1 copy drawn at turn X.

The above table summarizes the probability of the minimal at least 1 copy drawn at turn X (which is represented by rows) when you have Y copies of cards in your deck (which is each column).

#### How did I get the above table? (Read only if you are interested in actual statistics calculation)

As one of the reader asked how I got the table above, in this section I have decided to include about the actual part of the math I used.   This is pretty much standard in the world of card draw probability calculation.  It is called hypergeometric disribution.  In order to be able to use this function, two conditions must be met.

• The result of each draw can be classified into one or two categories.
• The probability of a success changes on each draw.

The actual formula becomes following:

$P(X=k) = {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}$

•  is the population size
•  is the number of success states in the population
•  is the number of draws
•  is the number of successes

In Excel or GoogleDoc, there is built in function hypergeometric.dist so that’s the function I used.  As far as the actual values I used are N = 60 i.e. deck size, K = # of copies, n = 7 + row # – 1 i.e. first turn you have 7 draw, second turn 8 and so on.  Then k = 0 i.e. you have drawn 0 copies of the target from the deck.  When you subtract this value from 1, it becomes the probability of AT LEAST 1 copy in your hand.

### 70% Cutoff

Now I picked 70% as target/reliable enough probability.  One may believe this number is a bit too small; whereas, the other may say I don’t need this high.  In those situations, just look at table above and find the corresponding turn.  But for now, let’s take 70% as winning 70% of games in TCG is amazingly high batting average.

First we need to understand the meaning of this 70%.  It means 7 out of 10 games you play, you will have at least one copy of the card you want by the turn X.  For the simplicity, based ont the table above, I created following table.

 Copies Turn 1 36 2 21 3 14 4 10

So this is basically saying if you include 1 copy of the card in your deck, you can only reliably assume that card be in your hand when you can get to turn 36.  Whereas if you include 4 copies, you can now reliably draw the card by turn  10.  This is a huge difference.  Most game won’t last until 36 turns, but many can last till 10 turns.

# Real world application

So coming back to the original question.  Is including 4 copies always better?  The answer is not necessary.  You have to apply what you learned from optimal resource card numbers article.

Let’s take a some specific example here.

### Step 1: I decided to make a deck revolves around a one single powerful card “The Kraken”.

This card has very strong ability; however, it comes with a hefty casting cost of 8 at its baseline.

### Step 2: Deciding # of resource cards in my deck.

Because this card has the cost of 8 to cast, I have to make sure there will be enough resources to meet the goal.   Perhaps this may be a bit too many, but I have decided to put 28 resources in my deck.  This will have the following probability distribution (see here for how I arrived this).

Based on the table above, I will have over 77% chance that I get 8 resources by turn 13 as long as I include 28 resources in my deck.

### Step 3: Deciding how many copies of Kraken to include.

 Copies Turn 1 36 2 21 3 14 4 10

Again, looking at the 70% copy table.  If I include 4 copies, I will have Kraken in my hand by turn 10.  But having Kraken in turn 10,11 or 12 won’t really help me much because there are still high enough chance that I may not have enough resources to cast the Kraken.

So now look at 3 copies.  It is 14 turns, which is right around exactly when I get enough resource to cast Kraken.  So rather than including 4 copies, I will include 3 copies of Kraken.

So again, why won’t I include 4 copies?

The reason is basically having a card that you cannot cast in your hand nothing but a form of card screw, in this case non-resource card screw.  You’d rather draw some other card that you can cast than one that you just hold in your hand for several turns without being able to use it.

## 70% each does not mean 70% total

The above analysis hopefully provided you some foundation for mathematical approach to how many copies of card to include in your deck.  But the specific example, I used in above have a flaw that is resource card draw & non-resource card draw in combination.  So I made it sound like 70% is your chance to draw Kraken and Play it based on the analysis, right?  Well actually not.  Since you have to have a both and each having 70% probability of happening in any game, it actually becomes 70 x 70 = ~50%.  (This is assuming two are independent event).   So despite the fact you optimized each, you still only have turn 13 Kraken play every other game.

This is why people especially beginner of TCG complain too much luck on TCG and you cannot draw a card you want or perhaps famous resource screw.  It is your deck design that essentially dictates 50% flip coin chance.  Obviously, if turn 13 Kraken play is so important you can include 4 Kraken, add card draw engine, and perhaps even 1 more resource card etc.

## Conclusion

Though there may be some flaw in this analysis, I hope this article provided a simplified way to conceptually understand why you include more or less  copies of a specific card in you deck, and including 4 copies or only 1 copy are not necessary better than the other.  They are situational, and remember probability speaks itself.  So you think you got a good deck and plan but never can get the right card combination at right time?  May be you need to look at statistics.