Twosomes or Threesomes in Yugioh

My Hanzo Harpie deck performed just well enough to goad me into tweaking it.  In the parlance of Yugioh, Dark Simorgh put in some work.  Perhaps, I could increase my chances of laying down the lock-down by adding a third big bird?  After all, my graveyard was consistently full of winds and darks.  

My first test hand consisted of two Ninjitsu Art of Transformaion, a Trap Stun, and 2 Dark Simorgh.  Yuck! 

After power shuffling, I tried again and drew 3 Dark Simorghs, 2  Harpie Hunting Ground, and a Fiendish Chain. Yuck²!  

While those hands are particularly bad, it did raise the question about when should you play three copies of a card over two? The most succinct answer is given by YugiTube’s TheRJB0.  He suggests the following algorithm: 

·         3 copies if the deck depends on it

·         2 copies for consistency

·         1 copy for tech 

It’s a reasonable rule of thumb, but what are the numbers behind this advice?  How much do we increase the chances of drawing 2 copies of a card when we add a third copy to our deck? 

Determining the percentage of specific hands in Yugioh (or any card game) requires a little knowledge of combinations.  A combination is the number of distinct subsets one can select from a defined set.  In this case, we want to know the number of six-card subsets (or hands) that can be formed from a 40 card set (or deck).  The order that the cards are drawn is inconsequential.  The answer to this question is given by the formula to the right.  Excel makes it easier by providing the COMBIN(n, r) function.  

In a 40 card deck, there are C(40,6) or 3,838, 380 six-card hands.  This will be our denominator when we determine our percentages. 

Next, we need to determine the number of hands that have duplicate cards if you play only two copies.  All of these hands will have the two copies (Simorgh, Simorgh) and any combination of four cards from the remaining 38.  Therefore, we can write this C(2,2) x C(38,4) or 73,815 hands.

 

The process is similar when determining the number of duplicate hands when you have three copies of a card.  In this case, you can have two cards of the three copies or C(3,2) and any combination of the remaining 37 cards or C(37,4).  You can also draw all three of your copies or C(3,3) and any combination of three cards from the remaining 37 cards or C(37,3).  Adding these numbers together gives us the total number of hands with two or three copies.  

Taking the number of hands with duplicates and dividing them by the total gives us our answer.  If you play two copies, you will draw into duplicates 1.92% of the time; if you play three, you will get duplicates 5.36% of the time. 

Let me put these numbers into some context.  Playing three copies means you will draw into duplicates once every 18.6 duels.  Playing two copies increases this number to once every 52.0 duels.  This difference may be significant since most regionals will generate 20 to 24 duels.  The risk of drawing into doubles has to be balanced with the risk of not drawing the card at all.  Adding a third copy increases your odds of drawing at least one copy from 28.1% to 39.4%.

In short, the great RJB0 is right.  Two copies double your chances of drawing the card without significantly increasing the risk of drawing duplicates.  With three copies, you better be ready to eat some crow.