In the manga Yugi holds off Kaiba’s three Blue-Eye’s White Dragons with Swords of Revealing Light for 3 turns in order to give him enough time to draw all 5 Exodia pieces and win.  Yugi does this in 10 turns and Exodia rules.  Or does it?  Exodia makes for great entertainment, but it’s not great if you want to beat your friend at the local tournament.  At the 2004 World Championships, 7 of the top 8 decks were based on Chaos.  None of these players even side decked Exodia.

This article is really about probability and I’m going to use Exodia as the example.  Just exactly what are the odds of repeating Yugi’s feat of drawing Exodia on or by the 10th turn?  Or what exactly are the odds of drawing Exodia for a First Turn Kill/Win (FTK)?  We all figure that the odds of drawing any given card are 1 in 40, but just exactly how true is that?  On the surface this makes sense and is seemingly logical since there are 40 cards in the deck and they are randomly shuffled.  It would seem that any given card has an equal chance of being in any of the 40 spots in your deck, right?  Well, yes and no.  Let’s look at the FTK first.

Let’s start by looking at a 40 card Exodia deck.  It is true that the odds of drawing an Exodia piece are 5 in 40 or 1 in 8 for the first draw at the beginning of a game.  So you have a 12.50% chance of drawing a piece and an 87.50% chance of drawing something else.  So it is true that you have an equal 1 in 40 chance of drawing any given card.  But, in the second draw we need to start thinking about how the math of probability really works.  You now have a 39 card deck and there is a 12.50% chance that there are only 4 Exodia pieces left and there is an 87.50% chance that there are 5 Exodia pieces left.  The odds are no longer 5 in 40; they are some combination of 5 in 39 and 4 in 39.  It is at this point at which the issue gets complicated.  If you work the math right you’ll find that for the first 2 cards drawn the probabilities are 76.282% that you’ll have no Exodia pieces, 22.436% that you’ll have one Exodia piece and 1.282% that you’ll have two Exodia pieces.  Think about it.  I worked this out in Excel using nothing more than addition, subtraction, multiplication and division.  I know that there are sophisticated ways that statistician folks work this out, but I got a C in statistics twice and I want folks to know you can do this yourself if you really apply some thought to it.  The key reason why the 1 in 40 odds don’t hold true beyond the first draw is that they don’t consider the impact of previous draws.

Ok, don’t hold your breath hoping for the Exodia FTK.  The odds are 1 in 109,668 that you’ll draw Exodia and win with the first 6 cards drawn from a 40 card deck.  In plain English this means that if you repeated the process of shuffling up a deck and drawing the top 6 cards 109,668 times, you ought to anticipate that you would draw Exodia once.  This suggests that the FTK with Exodia is EXTREMELY unlikely.  If you start out with Pot of Greed and then Graceful Charity you get to draw the first 10 cards off your deck in the first turn and you still have only 1 chance in 2,611 that you’ll FTK with Exodia; and I didn’t even weigh that down with the unlikelihood that you’ll get Pot of Greed and Graceful Charity like that.  Thanks to Ryusaki for doing a bit of research for me on Yugi’s 10 turn Exodia win over Kaiba in the manga (print version).  He was able to let me know what cards Yugi played, in order, by turn.  The probability of a making Exodia in 10 or fewer turns is much better than the odds of making Exodia in 1 turn.  In 10 turns you’ll draw 15 cards, thereby resulting in improved odds of 1 in 219.  Also thanks to Familiar Stranger who showed me that in the anime (TV version) Yugi wins with Exodia in the 14^th turn.  The odds of that are 1 in 328.  I suppose they needed to shorten the duel to make time for commercials.  So even in the two versions of Yugi’s duel the odds were still against Yugi, but that’s what the heart of the cards is all about.

Now the real point of the article wasn’t to try to get you to think about Exodia and how hard it is to make Exodia.  Nor am I looking to start a discussion about deck thinning, as I’m sure a number of you are thinking right now.  The real point is introducing the concept of real and calculated mathematical probabilities to those who play the game using Exodia as an example.  Like the Star Circle, it’s a tool that if used appropriately, can help you develop an advantage.  I also want to show that a 40 card deck is WAY better than a 45 card deck or even a 41 card deck.  You might think you know what happens to the above probabilities if you add 1 or 5 more cards, but do you really?

Take a moment to guess what the probabilities are for:

1)       Drawing 0, 1 or 2 Exodia Pieces in the first 2 cards drawn from a 41 and 45 card deck.

2)       Drawing Exodia FTK in the first 6 cards drawn from a 41 and 45 card deck.

3)       Drawing Exodia within the first 15 cards of a 41 or 45 card deck like Yugi did.

Check out the numbers as I’ve stated above for a 40 card deck and write down your best guess.  Do you think it’s just a little bit more unlikely to draw Exodia in each instance or a lot more unlikely?  Just how much of a difference can 1 or 5 more cards make?

After the second draw in a 45 card deck you will have 0, 1 and 2 Exodia pieces 78.79%, 20.20% and 1.01% of the time, respectively.  Similarly, in a 41 card deck you will have 0, 1 and 2 pieces 76.83%, 21.95% and 1.22% of the time, respectively.  So, there is little perceptible impact of adding more cards to your deck this early in the drawing process.

Things get a bit more interesting when we look at the Exodia FTK.  The odds jump from 1 in 109,668 to 1 in 203,626 in a 45 card deck.  A 41 card deck has better odds than a 45 card deck, but still significantly greater at 1 in 124,900.  As we extend the odds out further from the initial draw of the first card, the impact of adding cards becomes more magnified.  The impact of adding 5 more cards to your deck almost doubles the unlikelihood of drawing Exodia and Yugi would have a 1 in 392 chance of repeating the feat with a 45 card deck.  If you get nothing else out of this article, just remember this, it’s best to find ways to make your deck 40 cards, and no more.

I bet you’re wondering about the Excel spreadsheet I mentioned above.  If you want a copy, just e-mail me at notasperfectasyou@yahoo.com and I’ll send you the spreadsheet as an attachment.  I’d normally attach it to the article, but I want to see how many of you read it and took an interest in it.  If you request the spreadsheet you need to tell me what you thought of all four articles I’ve written.  Also, I’m not going to be giving Excel lessons.  If you don’t know how to work Excel, please find a friend to help you.  I’m sure the likelihood of your being able to figure this out is very good.

notasperfectasyou

It’s easy to find me on the message board, e-mail is ok, but I like open discussion better.

napay’s prior articles and why you’d want to read them:

“Fake/Counterfeit Yu-Gi-Oh cards” presents a personal experience of having bought fake cards and the lessons I learned from that experience.

“The Star Circle” presents a very visual way to think about the components of your deck and how to think about card flow as a way to improving your deck.

“How Big is your Monster” is about why you need to stop thinking about how to get big ATK monsters in your deck and why you do need to think about how cards work synergistically.
 

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