Should You Run Three Pot of Desires?
 Tom Viergutz
 20200625
There are different opinions on whether it is correct to run two or three copies of Pot of Desires in one's deck. The classic argument for running three is that "it is a very powerful card and I want to draw it as reliably as possible," and the classic argument for two is that "I never want to draw two copies."
In this article, I am going to give a simple overview of the problem with some basic probabilistic reasoning. I'm not going to give a conclusive answer, since I'll make some assumptions and ignore some points to make the argument simpler. But I will highlight things to consider that will hopefully help you when building your own deck.
The Scenario
Let's start with the assumptions I am going to make:

I don't care about the effect of Pot of Desires, I am just going to treat it as a card I want to draw one of, but not two. The arguments below will hold for any card that you are running duplicates of but don't want to draw duplicates of. If you want to do some further analysis, you might want to account for the fact that Pot of Desires also draws you more cards if it resolves.

I am going to think about the fivecard starting hand of a player going first with a 40card deck. If you are interested in thinking about other cases, it should be fairly easy to adapt the methods below to your own case.
Let's imagine now I am about to enter a tournament with my good pal Joshua Oosters. We have the same list, card for card, but on the day Josh can only find two of his secret rare copies of Pot of Desires! Since Josh is a giant rarity enthusiast, he refuses to play with lowerrarity Pot of Desires. He chooses to swap one out for another card that he wasn't originally running, let's say, Twin Twisters. I, having no such issues, happily run a deck using two secret rare and one common. Below, I will use the rarities to distinguish the different copies to tell apart different hands.
The Question
For now, we won't think about any numbers but just consider draws where my deck performs better against when Josh's deck performs better. We will compare side by side the hands I draw with the hands Josh would have drawn. A useful point at the beginning is we might as well I assume I draw my common Pot of Desires (and so Josh draws his Twin Twisters) since otherwise our hands would be identical anyway. So we only need to think about the four other cards in the hand where my fifth card is Pot of Desires, and Josh's fifth is Twin Twister. Let's look at the cases in a table:
Better for Tom  Better for Josh 

Neither of the secret rare Pot of Desires are in the other four cards. 


We are of course assuming here we would rather draw a lone Pot of Desires than a lone Twin Twister. Otherwise we would have put the Twin Twister in our deck in the first place! But I am comparatively worse off if we draw either of the secret rare Pot of Desires in our other four cards, since if we already have at least one then I would prefer the second to be Twin Twister instead. I have separated the cases of drawing two or three to make the probability easier.
This is all well and good, but as it stands it doesn't tell us very much. There are three key questions we need to answer here. Firstly, how often will these cases come up? Secondly, how much better are these cases for either me or Josh? Thirdly, how do we combine these facts to tell us what to do?
The Calculation
I can show you how to answer the first and third questions, but to answer the second is up to you. It will depend on the other cards in your hand, and more generally what deck you are running. Some decks can resolve two copies of Pot of Desires in a game, for instance, or can discard the second copy for a useful effect. This is a part you will have to figure out for yourself through your own testing and theory. This is also the bit you need to adapt if you want to apply this reasoning to different cards.
Answering the first question can be done with a geometric calculator online — or with some probability lessons from school. Many players have started using tools like this already. Remember we are assuming we have a fourcard hand with the fifth being Pot of Desires for me, and Twin Twisters for Josh. So our "sample size" will be four, our "population size" will be 39, and there are two "successes" in the deck.
(Here, the probabilities are what are called "conditional probabilities." I have assumed something has happened already, namely that I have drawn the common Pot of Desires. This is the easiest way to compare mine and Josh's decks. After all, if we only care about the difference between the two, there is no point thinking about all of the hands which would be the same for either of us. For the purposes of comparing which deck is better, what I have done here is totally correct. You will need to use different calculations if you want to work out how often you will draw two copies of Pot of Desires in any hand.)

Case 1
"Neither of the secret rare Pot of Desires are in the other four cards." We need the probability of exactly zero successes in our sample, which is 80.3%. So the vast majority of the time, I'll be drawing a better hand than Josh.

Case 2
"Exactly one of the secret rare Pot of Desires is in the other four cards." We need the probability of exactly one success, which is 18.9%.

Case 3
"Both of the secret rare Pot of Desires are in the other four cards." We need the probability of exactly two successes, which is 0.08%.
Now from this it looks like my deck is just better than Josh's, but remember, not all cases are equal. It might be that Case 1 is a little better for me, while Cases 2 and 3 are way worse. If we want to do some further calculations, we need to assign some numbers to just how much better each case is for me or Josh. What these numbers are out of is up to you — could be out of 1, 10, or 100 — so long as they are all on the same scale. Let's say D1 is how much better Case 1 is for me, and D2 and D3 respectively are how much better Cases 2 and 3 are for Josh. We expect my deck to draw better then on average if (by using a formula for what is called "expectation" for a random variable in probability) …
Josh's deck will draw better on average if the inequality is the other way around. Remember, the numbers D1, D2, and D3 you have to work out based on your own particular deck. From the looks of things though, unless D2 is significantly larger than D1, it looks like I was right to run three Pot of Desires. Better luck next time, Josh!
Opinions expressed in this article are those of the author and not necessarily Cardmarket.
6 Comments
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Excellent! (btw case 3 is 0. 008%)
Yesterday i went to a tournament with a crusadia deck and three copies. In my starthand there were two of them. The fact that i only had one of my fifteen crusadias in the starthand made me play both desires. In the end i banished 10 other crusadias just to draw veiler and my third desires
Always run 3 and play Magician's souls to pitch the dead ones
It is a neg 9 though
But all the joking aside, I think 2 Desires is right after all. The extra draw just isn't worth enough the times another copy of desires makes your hand worse if you play it at 3. Especially since it is very weak to handtraps like droll & lock bird or ash blossom. Moreover, if you brick with 2 desires in hand, you may need to use one of it first to even get your plays going, therefore going to risk your key cards getting banished.
I love Yugioh math.
For me it's 2 Desires in most of the cases, although it might be wrong. But it just feels better.
This article only mentions the likelyhood of opening with multiple Desires. Keep in mind that when playing 3 Desires not only does the probability to start with more than 1 Desire double, but so does the probability of drawing into Desire in case you only start with one in hand.
Haha math,
You can't draw 2 desires of 1 if you only play 2 which is why its better :)