PROBABILITY:

Probability measures the expected frequency in which an event will occur, of all possible

outcomes ( ). The probability of any event happening is designed to as or , where

is the event we want to calculate the probability of.

The formula is the following: where .0

The ‚#‛symbol means the cardinal of the Set, i.e. the number of elements in that Set.

BASIC NOTIONS:

Most time, we will be asked about the probabilities of multiple events happening. This can

either be by asking the probability of either event or event happening, for example, or

by asking the probability of both happening at the same time.

The first is noted . If both events are mutually exclusive (they cannot happen at

the same time), then it’s as simple as Otherwise, it’s .

The second is noted If both events are independent of one another (the fact that

one happens does not inherently affect the other), then we calculate

OPPOSITE EVENTS:

Sometimes, it’s easier to calculate than , where is the opposite event to

For example, would we be flipping a coin and the event , then .

Note that they are mutually exclusive, so

This last formula can be explained by saying that and form a partition of the total ( ). As they are total opposites, there’s no in-between: Either happens, or happens.

FACTORIAL NUMBERS:

Combinatorics studies the possible combinations of the elements within a Set.

We will start by defining the factorial operation, also noted as

That means we multiply by all natural numbers that are smaller than itself, until 1.

By convention, .

1) Probability - Most Fundamental Basics

2) Combinatorics - Basics

COMBINATIONS OF A SET:

To calculate all possible different combinations of any given Set, the formula to use is:

Here are some examples of Sets, and how many possible combinations they have.

SET 1:

SET 2:

SET 3:

SET 4:

SET 5:

The number of successes in drawing cards we want from a Deck, follows the Hyper-

Geometric Distribution, but since that’s a little more complicated than what I’m aiming for,

I will calculate the probabilities the ‚easy‛ way.

Let’s say we want to calculate where the event ‚Drawing at least 1 copy of an

Unlimited card in the 1st turn of the Duel‛. If we designate by the number of copies we

want to draw out of all possible 3, we are calculating the probability of .

Since calculating is rather difficult, I will instead calculate . Since and we

can’t take negative values, we have that :

The first step can be explained by seeing that:

The calculations done are easy to explain. We don’t want to draw any copy of our card in

, so we want to draw any of the other 37 cards in our 40-card Deck. With each

draw, the number of cards decreases in both numerator and denominator.

Finally, we must multiply by all possible combinations in which these cards came to us.

In this case, we have 6! / (6! · 0!) = 1. Most people don’t do this, but it’s a necessary step!

We are going to practice a little more.

3) Calculating Probabilities for drawing cards in Yu-Gi-Oh!

This time with ‚Drawing exactly 1 copy of an Unlimited card in the 1st turn of the Duel‛.

We want to draw 1 copy of our card, then we want to draw any of the other cards in our

40-card Deck. With each draw, the number of cards decreases.

Finally, we must multiply by all possible combinations. In this case, 6! / (5! · 1!) = 6.

‚Drawing at least 1 copy of a Semi-Limited card in the first 5 turns‛.

We don’t want to draw any copy of our card in , so we want to draw any of the

other 38 cards in our 40-card Deck. With each draw, the number of cards decreases.

Finally, we must multiply by all possible combinations. In this case, 10! / (10! · 0!) = 1.

Should have been to draw exactly 1 copy, then it would have been 5! / (4! · 1!) = 5.

‚Drawing all Exodia parts as the starting hand‛. Let be the number of successes in

drawing the cards we are tracking. In this case, we want 5 card(s) out of 5.

‚Drawing exactly 2 Tribute Monsters in the 1st turn, if we have 4 in the Deck‛. Let be

the number of successes in drawing those cards. In this case, 2 card(s) out of 4.

It’s not that difficult once you get used to it, as you can see.

Having covered all the basics, and having taught you how to calculate the probabilities for

yourselves, I will go on to discussing the first false belief deeply rooted on this game.

The most controversial will be the first: The all-powerful sacred 40 card-Deck.

Most people blindly believe that any number besides 40 for a decksize in Yu-Gi-Oh is just a

terrible mistake at Deck making. I’m sorry to put it like that but… well, that’s blatantly

stupid. Such extremist opinions give nothing to the game, and instead take away from it.

Of course the probability of drawing at least 1 card out of ‘m’ cards (with in a

40-card Deck is higher than in any other legal decksize, that’s obvious. Nevertheless, that

isn’t all we have to bear in mind.

Having a smaller Deck significantly increases the probability of drawing multiples of each

card. While a lot of people would not give a 2nd thought on the matter, thinking that such

an event is rather beneficial, I will stop you right there and make you think a little.

Some cards are incredible when drawn at 1, but become a nuisance if extra copies start to

stack on the hand. Also, by having higher probability of drawing extra cards of any group

of cards in your Deck (some people think that this argument only applies to copies of a

card, but it extends to any group of cards in your Deck), you can end up with a hand full of

cards with similar effects, while you are waiting for any other card to come by. What’s the

use in drawing more of the same, if you don’t get a chance at playing anything else?

EFFICIENCY FACTOR:

Efficiency is a function that expects 4 parameters: the decksize of the New Deck, the

number of cards we are tracking in both the New Deck and the Reference Deck (40-card),

the number of successes (cards we want to draw out of ), and the number of draws.

The result is a number in the interval , which tells you how much more probable is

an event in the New Deck than in the 40-card Reference Deck.

BASIC PROPERTIES (this is explained in-text, and so it’s not necessary to understand):

1) For fixed and , increasing the value of will increase Efficiency.

2) For fixed and , increasing the value of will decrease Efficiency.

3) For fixed and , with , if increases, Efficiency will increase to +

4) For fixed and , with , varying within [m, 40] will not change Efficiency.

From now on, I’m going to use the same value for , and contrast the new decksize with

the traditional widespread 40. In particular, I will use 44 (decksize I normally use).

Let’s do some calculations.

4) 1st Fallacy of Yu-Gi-Oh! TCG : The 40-card Deck

Drawing exactly 1 copy of an Unlimited card in the 1st turn of the Duel‛.

Therefore, we have that a 44-card Deck is closely efficient to delivering us 1 copy of an

Unlimited card compared to the Reference Deck, with just a loss of ~0.07 in Efficiency, and

increasing Efficiency with each draw (Property 3).

Drawing exactly 1 copy of a Semi-Limited card in the 1st turn of the Duel‛.

Drawing a Limited card in the 1st turn of the Duel‛.

Here the difference is the highest at ~0.09, but sadly it won’t increase further like in the

cases before, because of Property 4 ( ). This makes it the most resentful for increasing

the Decksize above 40.

a

What does this all mean? That the loss of probability in drawing a copy of a card from a 40-

card Deck to a 44-card Deck is so small, it could be considered almost negligible. The New

Deck is still very efficient at delivering you these cards, with a loss that doesn’t get to 0.1,

and it further increases with each draw -- except when trying to draw a Limited card.

On the other hand, let’s suppose we are now tracking a group of 8 cards in our Deck.

Drawing exactly 1 card out of 8 in the first 10 turns of the Duel‛.

As we can see, now it results that for drawing exactly 1 card out of 8, our 44-card Deck is

vastly more efficient than the Reference Deck, with a gain of ~0.4

However, one may argue that we might want more than just 1 card out of 8. Let’s see that

the 44-card Deck is still more efficient at delivering us 2 cards out of 8 than the standard 40.

Drawing exactly 2 cards out of 8 in the first 10 turns of the Duel‛.

draws

draws

Let’s also calculate this one:

Drawing exactly 1 copy of an Unlimited card in a Duel lasting 15 turns‛.

Which means that it’s more probable to see exactly 1 copy of any Unlimited card in those

15 turns in a 44-card Deck than in the Reference Deck. What’s happening here? How can it

be the probability is higher in a Deck that is not a 40-card Deck?!

Well, you can check all the calculations. If you did right, you would see they are all correct.

What is happening here, is that the probability of drawing exactly X cards out of Y is not

cumulative, and graphics a curve. This is the probability for drawing exactly 2 cards out of

8 in the Reference Deck:

The higher the higher the probability of drawing a few of those cards compared to the

Reference Deck, sometimes even a lot higher, but the probability of drawing more cards is

reduced quicker. (Properties 1 and 2 of the Efficiency function)

CONCLUSIONS:

Any X-card Deck, with , is more efficient at drawing a few cards out of card

groups than a 40-card Deck, but is specially far more efficient at not clogging the hand up

with cards of the same group, thus making your hands more varied.

It all comes at a trade-off, of course. The installed mentality of using just a 40-card Deck

does not come without its reasons. The larger the Deck, the less efficient at drawing copies

of any card. However, for a X-card Deck is closely efficient to a 40-card Deck

in drawing copies of any card, with a negligible loss on most cases (except Limited cards).

I hope this clarified many things, and lets Duelists to get out of the restrictive 40-card-only

Deck mentality, and consider more factors into deciding their Decksize in the future.

The higher , the lesser the probability past the maximum of the curve. Why? Because by

then you’re expected to draw more cards out of those you are tracking, and therefore

the probability of drawing less than the new expected values goes down.

Now, I’ll delve into the realm of randomness.

To achieve proper randomness in a non-virtual Deck, we use different shuffling techniques

in real life. There are basically four groups when it comes to them:

1) Riffle or Weave/Faro Shuffling.

They are based on separating the Deck on two halves, and then intertwining them. Any

shuffling move that does not change the top and bottom cards is an out-shuffle, otherwise

it’s an in-shuffle. Note that barring error done by hand, even if combining at random in

and out shuffles, this is a deterministic technique and the results can therefore be calculated

and known beforehand.

2) Overhand or Hindu/Kattar/Kenchi Shuffling.

They are based on taking groups of cards at random from the Deck, and replacing them

elsewhere at random too. It’s truly random, but inefficient as it takes way too long to truly

randomize the Deck without leaving clumps of grouped cards.

3) Pile Shuffling (also known as Power Shuffling).

It is based on making X piles of cards from taking the cards on top of the Deck. This kind of

shuffling breaks clumps of cards, but does not randomize a Deck, but just re-orders it in a

deterministic fashion, even if the piles are re-arranged at random (though then it becomes

arguably impossible to determine the outcome if several iterations are performed).

4) Irish/Wash Shuffling.

Scrapping the cards on a surface, stirring them, and finally taking them one by one. If the

cards are gathered again following a non-random fashion, it can be argued that the result is

not random. Also, the distribution of the fallen cards follows a Poison Distribution, and is

therefore somewhat non-random too.

Programming them is rather easy, though not a matter I would share, as most won’t

understand or even be interested in. The code in which I programmed them in Wolfram

Mathematica is still available for anyone who wants it, and I will post it on demand.

It suffices to say that Riffle Shuffling cannot be emulated perfectly, as when done by hand,

the intertwining is not perfect, as handmade error happens. I cannot emulate this error

properly, and so I have programmed the function to do perfect shuffles instead. Whether it

is an in-shuffle or an out-shuffle is decided at random, to maximaze randomness.

Coding the Irish/Wash Shuffling is also difficult, as there’s no way one can emulate it,

knowing that in real life a certain pattern is followed, but the gathering of the cards can be

totally different from one person to another. Because of this, it hasn’t been programmed or

considered at all. It is also rather time-inefficient, so no point in discussing it further.

5) An overview at Shuffling Techniques and Programming them

Of course, programming the different shuffling techniques is not for leisure, but to make an

experiment and decide which are the best methods for properly randomizing a Deck.

The definition of true randomness would be assigning each card a unique number, and

then having the numbers to be taken randomly. This can be done by Pile Shuffling in

piles, where is the size of your Deck, then re-arranging the piles at random.

Using the shuffling techniques on an already truly randomized Deck will yield no results:

it won’t randomize the Deck further (if that makes any sense), nor de-randomize it. For that

very reason, we will only use the different coded shuffling techniques on Sorted Decks.

That means the Deck is ordered in blocks of Monsters/Spells/Traps.

However, to properly set an experiment to compare the different results, we must first

choose the Statistics that we will use. The classical statistics may not give us enough

information for randomness, and even less in such a complex game as Yu-Gi-Oh is.

Therefore, we have to choose appropriate Statistics. These are the two I will use:

Clumps Found

We dichotomize the Deck into 2 categories: Monster cards and non-Monster cards. Further-

more, we suppose those categories follow a 1:1 ratio. Each time we find a card of a category

next to another card of the same category, it counts as 1 Clump Found.

A Sorted Deck with a 1:1 ratio of Monsters vs. non-Monster cards, has pair cardinality and

yields exactly - 2 Clumps (if the cardinality is odd, then results are less reliable). That

means that a truly randomized Deck should have approx. clumps found.

Seeker Card Distance

We number the cards in the way they are ordered, and then get to them in the order they

are numbered after Shuffling. IN DETAIL: For numbered card X, we seek the next card

(X+1), and write the distance between them (counted in cards). We add all written numbers

to a grand total, which is the SCD (Seeker Card Distance).

This is a very sensitive measure for randomness, but it takes longer to compute, and so is

used only when more precision is required.

A Sorted Deck yields exactly - 1, because the are cards separated 1 card afar each. The

last card does not seek any next card, so only - 1 cards matter.

SCD varies a lot through randomized Decks, being an average calculated upon each Deck’s

decksize ( ) the best option.

6) Preparing the Experiment – Choosing the appropriate Statistics

Riffle or Weave/Faro Shuffling

Let’s start by testing the randomness of our coded Riffle Shuffling. Remember it does only

perfect shufflings, alternating between in-shuffles and out-shuffles at random (with equal

chances - 50%) at each iteration. I will test it on both a 40-card Deck and a 44-card Deck.

Clumps Found on a 40-card Deck. Clumps Found on a 44-card Deck.

The X-Axis are the number of times the Riffle Shuffling algorithm is applied, while we have

on the Y-Axis the proportion of Clumps Found (out of the max. - 2).

Obviously, one perfect Riffle Shuffling evens the cards out in a pair-cardinal Deck, and so it

finds no clumps. However, starting from the 2nd Riffle onwards, we can see how far from

the 0.5 mark it goes, stabilizing slowly as the number of iterations grow.

Expecting the error done by hand to change a few cards here and there, we can still take a

low number of iterations in these graphics to be pretty accurate to real life results, and

those are totally busted and non-random at all.

Let’s get now to our other statistic, Seeker Card Distance (SCD). By using a true random

shuffling technique (Power Shuffling into piles for a -card Deck, and gathering the piles

at random) in 4000 different tries, we are given the statistic’s average for each decksize.

SCD on a 40-card Deck. SCD on a 44-card Deck.

AVERAGE: AVERAGE:

7) 2nd Fallacy of Yu-Gi-Oh! TCG : Shuffling Techniques

The X-Axis are the number of times the Riffle Shuffling algorithm is applied, while on the

Y-Axis is the Seeker Card Distance, where the green line indicates each graphic’s SCD

average for their decksize, calculated as said above.

As we can see, the values are rather far from those green lines most of the times, and only

get stable and nearer the marked average when the number of iterations gets close to 8.

CONCLUSION: Riffle Shuffling is not fit for randomizing a Sorted Deck, unless you do 8+

iterations. Fewer iterations totally DO NOT randomize a Deck, and can give awful results.

Addendum: There is a demonstrated formula for determining the number of minimum ite-

rations for truly randomizing a N-card Deck, it’s: . It yields approximately 8.

Pile Shuffling (also known as Power Shuffling)

This method of shuffling can be used for self-advantage, even when not stacking the Deck.

What do I mean with that?

Let’s consider a -card Deck, and now let’s suppose we Pile Shuffle into piles, then into

piles. Moreover, let’s suppose we gather the piles in any order we want (even randomly)

each time. What’s the result? In terms of Clumps Found, our 1st statistic, this is the result:

Case 1) There are exactly Clumps Found.

Case 2) There are at maximum Clumps Found.

As you see, that means that Power Shuffling can be used to consistently even out the cards

in a Deck in a non-random fashion, following a desired Monster/Spell-Trap/Monster/Spell-

Trap/Monster/Spell-Trap... distribution after shuffling, which is cheating to an extent.

[The results posted here have been tested both in real life, and in computer simulations in

4000 different tries. You can check for yourself, though, if you are skeptical.]

Because Pile Shuffling is totally deterministic, and it can be cheated for advantage even

when not purposely stacking the Deck, I see no point in researching it further for a legit

and useful shuffling technique, unless used solely for Clump breaking.

Overhand or Hindu/Kattar/Kenchi Shuffling

The method of distributing cards in Overhand Shuffling is translocating groups of cards

from place to place, so it's expected that its effectiveness in a Sorted Deck to be subpar

comparing it Riffle/Weave/Faro Shuffling, even when this method is truly random, unlike

the other which is more deterministic and needs to do a determinate number of shuffles to

get close to true randomness.

We will be taking values for from 1 to 5 (number of passes, which includes 6 group

exchanges per pass, with varying cardinality from as low as 3 and as great as 1/3 Deck).

Clumps Found on a 40-card Deck. Clumps Found on a 44-card Deck.

The X-Axis are the number of times the Overhand Shuffling algorithm is applied, while we

have on the Y-Axis the proportion of Clumps Found (out of the max. - 2).

SCD on a 40-card Deck. SCD on a 44-card Deck.

AVERAGE: AVERAGE:

The X-Axis are the number of times the Overhand Shuffling algorithm is applied, while on

the Y-Axis is the Seeker Card Distance, where the green line indicates each graphic’s SCD

average for their decksize. Contrary to Riffle Shuffling, it performs a logarithmic curve.

CONCLUSION: For effectively shuffling a Sorted Deck we need to do at least 12 passes of

a thorough Overhand/Hindu/Kattar/Kenchi Shuffle. Although it's less efficient than Riffle

Shuffling as it requires slightly more iterations, its results are closer to true randomness

than those of Riffle Shuffling.

Pile Shuffling is totally out of the question since it can be used for cheating (although it's

the best method for reducing drastically the clumps found on the Deck, and if shuffled

afterwards or before other techniques, the Deck will still be random at the end).

FALSE BELIEF DEFEATED: Riffle Shuffling is more pro, and is overall better than any

other shuffling technique for properly randomizing a Deck.

Overhand Shuffling is better for just 50% more minimum iterations, and is truly random.

FALSE BELIEF 1: All decksizes apart from 40 are inherently worse, as 40 is the best

decksize one can choose. Having extra cards is a mistake that has to be avoided.

False. Why?! Because it restricts the variety of cards that will appear in one’s hand, which

makes drawing duplicates of cards -or cards with similar effects- much more common.

FALSE BELIEF 2: Riffle Shuffling is the best method for properly randomizing a Deck, and

all other shuffling techniques are not as good.

False. Why?! Riffle Shuffling is deterministic, and 8+ consecutive shuffles are needed for

getting decent randomness. Overhand Shuffling on the other hand, while less efficient,

gives better results. Plus, it cannot damage the cards in the process, and is truly random.

ALSO OF INTEREST:

pages 2-3: Teaches how to properly calculate probabilities in Yu-Gi-Oh.

page 10: Interesting and shocking result about how Power Shuffling can cheat in the most

innocent and unpredicted ways.

CLOSING THOUGHTS

I really hope that after all the read, the Yu-Gi-Oh community gets unnatached of those two

deeply rooted beliefs, and start expanding their horizons and thinking freelier. I would also

hope for a better understanding and use of Mathematics, but alas, miracles don’t exist.

I am truly grateful for you, reader, for lending me some time to read this.

I wish you the best in your dueling.

NOT AGREEING?

If you find any discrepancy, and have some Math knowledge to back you up, don’t hesitate

and voice yourself on the thread!

8) Conclusions and Closing Thoughts