The Mathematics of Pot of Desires

has been the cause of many debates among Yu-Gi-Oh! Players in the past. In this article, I will explain the math behind both the argument for and against running zero, two, or three copies of Pot of Desires to help you decide how many copies your deck(s) should play.

Pot of Desires

Ever since the original was banned, many new have been released. Most of these pots include Pot of Greed's infamous "draw two cards" effect or some other way to add cards from your deck to your hand. But Konami wasn't about to just release a bunch of new Pot of Greeds into the format, as the card was banned for a good reason. So they added restrictions to subsequent pots. In his recent article, my colleague Sam outlined the history of draw-two cards in Yu-Gi-Oh! and mentioned how much people are willing to pay to draw two cards in Yu-Gi-Oh! Pot of Desires is no exception here. For the cost of banishing ten cards from the top of your deck face-down, it allows you to draw two cards.

People frequently argue about whether you should run two or three copies of Pot of Desires in your deck. Others even recommend not running it at all because of the high risk associated with its activation cost. Going forward, I'll show the probability of various scenarios involving Pot of Desires. The goal is that by the end of the article, you will know the mathematical arguments for running zero, two, or three copies of Pot of Desires. Pot of Desires is currently unlimited, so I don't really see any reason you would choose only to run a single copy of the card.

Opening Hand with at Least One Copy

To start the discussion, we need to check on the probability of having Pot of Desires in your opening-hand. For all the calculations in this article, I am assuming that you are playing exactly 40 cards in your deck. I've already explained the specific formulas you need to calculate these probabilities in a previous article. If you run two copies of Pot of Desires in your deck, the chances of opening at least one copy are about 23.7% when starting first and 28.1% when you start second. In a deck with three copies of Pot of Desires, you would have at least one copy in your opening hand in 33.8% of our games going first and 39.4% when going second.

You may notice that the probability of opening Pot of Desires is about 10% higher (in absolute terms) when you run three copies instead of two. This is a rather large difference, clearly. Therefore, if you want to see Pot of Desires in your opening hand every game, it's very important that you opt to play three copies in your deck. Please note that these probabilities are valid for any card in the game. You can apply the same argument when you contemplate whether to play two or three copies of in your deck.

Opening Multiple Copies

But when playing more than one of a card, you must also consider how often you'll open more than one copy of Pot of Desires. One of the major arguments for playing two instead of three copies is that the increased chance of opening multiple copies of the Pot. So, let's look at those probabilities.

In a deck with only two copies of Pot of Desires, there is only one way to open multiple copies, namely drawing all copies in the deck. The probability of opening both copies of Pot of Desires in a deck running two copies is about 1.3% without the extra draw and 1.9% with it. If you play three Desires, you will open two or more copies in 3.6% of all games going first and 5.4% of games going second. Thus, the probability of opening multiple copies of Pot of Desires almost triples when you play three copies instead of two copies.

This increase seems huge at a first glance and it might suggest that you're better off playing only two copies. However, the probabilities are both quite low overall. In a ten-round YCS, assuming you play three games in each match, you would play 30 games in Swiss rounds at the event. If you play first in half of those games and second in the other half, you will open multiple copies in 1.4 games out of the 30 played, assuming you play three copies. Realistically, you wouldn't play 30 games, so you're looking at an average of one-two games where you open multiple copies over the course of the tournament. In this light, the case for running fewer copies weakens considerably.

Drawing Desires Off Desires

Another argument for running two copies refers to the probability of drawing another copy of Pot of Desires with your Pot of Desires. Once again, we will compare a deck with two copies to a deck with three copies. For simplicity, we will assume that you have only opened a single copy and all other copies are still in the deck. We'll look at the two-copy case first. After drawing your opening hand, you would have 35 cards left in your deck. One of them is your second copy of . When you activate the Desires in your hand and banish ten cards from the top of your deck face-down, there are two possible outcomes. If your second copy of Desires Is banished face down, then there is no chance of drawing the second copy. This happens 28.6% of the time. In the 71.4% of all cases that you don't exile your second copy face down, then the probability of drawing your second copy from the remaining 25 cards is 8%. Overall, you will draw your second copy in about 5.7% of all cases on average.

It becomes a little trickier when we consider the case of three copies of Desires. In this case, we must differentiate between banishing zero, one, or two of the remaining copies with the first copy. The probability that you banish none of the excess copies is 50.4%. In this case, you would draw at least one copy of Desires 15.7% of the time on average. In this case, you would draw both extra copies of desires 0.3% of the time.

What happens then when you banish one or two copies of Pot of Desires with your initial Desires, which happens 42% and 7.6% of the time respectively? As before, if you end up banishing both remaining copies, you have no chance of drawing a copy. If you banished only one copy, then the chance of drawing the second, also as before, is 8%. Combining all these probabilities yields a total probability of 11.3% to draw another copy of Desires with Desires when it is the first card you activate and you have two more copies left in the deck.

This is an increase from 5.7% to 11.3% when running three instead of two copies. These calculations were all done assuming that Pot of Desires is the first card you activated in the game when there were still 35 cards left in your deck. If you did the same calculations again with only 30 cards left in the deck, you would find that the corresponding probabilities are 6.7% and 13.1%, respectively. From this, it's reasonable to conclude that drawing into another copy of Desires off your activation could be an issue in several games over the course of a tournament – more specifically four out of 30 games from our previous example.

Admittedly, this is a much bigger drawback. The chance of drawing your second or third desires with your first goes up significantly when you choose to run the extra copy. While this is a clear drawback, you do still go +1 in card advantage whenever you activate Desires, even if you draw an extra copy. The probability of drawing two copies is, additionally, so low that it's not worth considering.

As a side note, I want to mention that there is a psychological component to this discussion. We tend to remember negative events more commonly than positive ones. Negative events also tend to affect our psychology more than positive ones. As an example, the majority of Yu-Gi-Oh! players would most likely be more upset about their starlight rare being stolen from them than they would be happy to pull a starlight rare Lightning Storm from a booster pack. Even though the win/loss ratio is the same, the negative event would likely have a higher impact. Similarly, we tend to remember those games in which we drew Desires off Desires and lost the game because of it better than those in which it actually netted us two new cards and we won the game. It's also much harder to know whether you would have drawn that Desires that won you the game with one less copy than it is to know when drawing Desires of Desires kills you.

Banishing Combo Pieces

Another argument against is the risk of banishing combo pieces. I will now refute, or at least weaken, said the argument by showing you the probabilities of actually banishing all copies of a combo piece in a deck. To simplify things, we will assume that you haven't drawn your combo piece yet, implying that all its copies are still in the deck. We will also still perform these calculations under the assumption that you activate Pot of Desires with 35 cards left in your deck. Realistically, this may not be the case so the probabilities might be a bit higher if you activate Pot with fewer cards in your deck. Our calculations will, however, provide us with a valid lower bound for this circumstance.

If you are only playing one copy of a combo piece, the probability that you will banish it with Pot of Desires is 28.6%. If there were only 30 cards left in the deck, the probability would be 33.3%. Thus, in over a quarter of your games, you would banish your only copy of the given card with Desires. It's unlikely, however, that this card would be essential to your combo if you're only running a single copy of it, unless it is searchable through other means.

If you play two copies of the combo piece, you banish both copies with Pot of Desires 7.6% of the time. While most important cards are played in threes, there are cases where two copies are actually the correct number, so this case is more important. For example, is vital for recursion strategy. However, you only need one Jack Jaguar for this in your graveyard. Running two copies of it will shrink the probability of losing all copies of it from 28.6% down to 7.6%.

Lastly, the probability to banish all the copies of a combo piece with three copies in your deck is a mere 1.8%. Very few games would lead to you banishing all the copies of an important combo piece, especially if you play three copies in your deck. It's reasonable to conclude that, even though there is a risk of banishing all the copies of a given card, that risk is fairly small and well worth taking. Even though it might hurt to banish all copies of a given card, very few decks play essential combo pieces that can't be searched out. Furthermore, the probability of not drawing a three-of within the first 10 cards is 41.1%. You have a higher probability of just not drawing your unsearchable three-of. So you may as well run the card that helps you find it more often.

I have summarized most of the above probabilities in the following two tables. The probabilities preceded by a ≥ in both tables are only lower bounds since for all events it was considered that there were still 35 cards left in the deck when Desires is activated.

To summarize, I have demonstrated that running three copies as opposed to two greatly increases the chance of seeing it in your opening hand. The added risk from this is comparatively small. The most convincing argument against running three copies comes from the potential of drawing into an extra copy of Desires. The probability of this does increase significantly when running three as opposed to two copies. The probability of banishing important combo pieces is negligible and not worth considering. What are your thoughts on Pot of Desires? Do you run two or three copies in your deck? What are your motivations for doing so? Please let me know your thoughts down in the comments! Did you notice which “Pot of” spell card was not featured as a picture in this article?

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