 # Demystifying Probability

Probability is undeniably important to deck-building as well as playing. This article aims to provide the tools needed to start making key calculations.

You may be relieved to read that this article will not go into the maths itself. Instead, all of the calculations today will be done by the new Yugioh Deck Probability Calculator, created by Rustywolf and I!

Let’s see an example of its use. Suppose we want to find the probability of opening at least 1 Edea or Eidos (and we run 3 of each), as well as at least 1 Monarch out of 10. We can say we have a 39 card deck due to Upstart Goblin, and if we’re starting first, we have a hand of 5 cards.

Let’s put all of this into the calculator. (Handy tip: you can use the up and down arrow keys on your keyboard to change the numbers in the boxes as well).

http://prnt.sc/b701hf

44% means that we can expect to draw this combination roughly 44 times out of 100 games.

If we want to work out exactly how many games a probability corresponds to, we can just ask Google. If we type in “44% of 25” it calculates it to be 11 games out of 25 (which is roughly the number of games in a 10 round tournament).

Now, probability is not exact. It only tells us what to expect over a large number of games. We might go through 10 games and never draw the above combo, or we might get the above combo 10 games in a row. But this should happen very rarely, and if we were to play a large enough number of games, we’d see that the number of times we draw the combo will line up closely to what the odds indicate.

Let’s see another example. Suppose we’re wondering whether it’s worth running Allure in a 40 card deck with 5 DARK monsters. It’d be good to know how often it’s going to be dead.

Let’s put it into the calculator. Allure is dead when we draw 1 Allure but 0 Darks:

http://prnt.sc/b704f9

7%. How does that compare to the number of times it’ll be live?

http://prnt.sc/b704w6

This confirms what we might have suspected: with only 5 Darks, Allure will be dead more often that it will be live (in the opening hand). This suggests that we should either cut Allure or try running more Darks. We can also use the calculator to modify the number of Darks we have in the deck and explore how the odds change.

Now let’s take a look at a more complicated situation. Suppose we want to know the probability of opening both an Upper scale and a Lower scale in our (or our opponent’s!) 40-card Performapal deck.

Let’s say we have:

10 Lower scales
7 Upper scales
3 Performapal Skullcrobat Joker

Performapal Skullcrobat Joker counts as either scale, since he can search either on a Normal Summon. (We will also assume for this example he could be placed as a scale himself if needed, so if we get two of them, we can set one of them and search a Lower scale with the other).

Joker complicates things. We cannot simply add 3 to both the number of Lower and Upper scales to account for him. Instead, we have to think a little about the possible ways there are to get both a lower and upper scale.

Case 1: We draw no Jokers, at least 1 Upper scale, and at least 1 Lower scale.

Case 2: We draw exactly 1 Joker, and at least 1 Upper or Lower scale

Case 3: We draw 2 or more Jokers.

We can now calculate each of these probabilities individually, and then add them together to get our final answer.

Note that the cases above don’t overlap, because they all have different numbers of Jokers in them. If they did overlap, we couldn’t add the probabilities to get a sensible answer.

Case 2: http://prnt.sc/b9ytt7

Case 3: http://prnt.sc/b9yr8t

Adding them up, we get our answer of 66.22% (which, incidentally is close to 2 out of 3, an important threshold when playing a match)

The moral of this situation is that probability can be quite delicate. We need to think carefully about the hands we want and split them into cases we can use the calculator for. And call me crazy, but it’s actually quite fun. It’s maths, but it’s got a very Yugioh feel to it, because you’re thinking carefully about what kind of hands you want to draw and what the best way to categorise them is.

There are a few techniques that can be used to troubleshoot . Here are some questions I like to ask myself when calculating things, which lets me catch problems with the maths or its use before I move on with my calculations.

Do my cases overlap?
The above scenario was an example of a case where we had to be careful with this.

Have I missed out any possibilities?
As another example, one might calculate the odds of drawing exactly 1 Wind-Up Magician and exactly 1 Wind-Up Shark (back in the Wind-Up format), and get a figure of 11.6%. However, a hand of at least 1 Magician and at least 1 Shark can combo just as well.

http://prnt.sc/b706bf

The odds of drawing at least 1 Magician and at least 1 Shark is 13.90%.

Have I counted too many possibilities?
In the above example, it turns out we need Wind-Up Magician in the Deck for the Wind-Up Combo to work. So we actually need to limit the number of Magicians we want to draw down to 2 instead of 3.

http://prnt.sc/b70660

The chance of drawing what we want ends up dropping slightly to 13.86%.

In this example that doesn’t make a big difference (0.04% corresponds to 4 games in every 10,000) but in other articles we will see more calculations in which we need to be wary of overcounting.

Might my opponent factor into things?
You might want to factor in vulnerability to the opponent’s traps and hand traps.

For example, one can calculate the odds of an opponent drawing into at least 1 of their 3 sided Ghost Reaper and Winter Cherries or 3 Maxx “C”:

http://prnt.sc/b9zi52

Yikes, that’s more than half the time. If you’re a BA player, this is something important to be aware of.

In general a game isn’t just decided by whether you draw a single combo in the opening hand or not, and most games will span several turns, in which time you will draw lots of cards and thin your deck out. This means that if you’re looking for the chance of drawing into a combo at some point in the game, then the odds could be considerably higher than they are for just opening the combo.

But the maths becomes rough and complicated very quickly, as the possibilities explode due to the sheer number of different moves that you and your opponent can make. The more we don’t take into account, the less accurate and reliable our calculations become.

Quiz time!
Use the calculator (or don’t!) to answer the following questions.

(1) Say you have a 40 card deck and a 6 card hand, and you run 3 copies of MST. What is the probability of opening at least 1 copy in the first turn? Roughly how often is this?

(2) How do the above answers change for the starting player now that he only opens with 5 cards? (You may also replace “MST” with “Twin Twister” if you like.)

(3) In how many games of an 8 round tournament would you expect to open MST (or Twin Twister) if you always go second (so your opening hand is 6 cards)?

(4) In a 40 card deck and a 6 card opening hand, what is the chance of drawing at least 1 card out of 8 cards? Is this a good figure? (This is known as Jordan’s 8/40 Rule).

(5) How does the above answer change for a 5 card opening hand? Is Jordan’s 8/40 Rule still useful?