Ash Blossom is not a short print (DEMONSTRATION)

Hand traps are both torment and delight of players in the current meta. This is why the news of the reprint of Ash Blossom & Joyous Spring (or shortly Ash) made lot of people elated and raised hopes of a lower price on the secondary market. An initial lowering had been present, actually, until when a rumor saying that Ash was short printed spreaded, since in 12 opened box, usually yugitubers & co. found around 3 or 4 of the card (so 1 every "a bit less of" 4). Is the rumor actually true? With the help of a bit of easy mathematics, we are going to demonstrate that the pull rateo aren’t that weird and in line with what Konami asserted many times: Ash Blossom isn't short printed.



2 plus 2 is 4 minus 1 is 3 quick math
One of the most known definition of probability (and the most used outside the universities) is:

 

Or rather, the number of events in which the condition is met, divided by the total possible events. We will obtain a number between 0 and 1, and multiplying it by 100, we will have the percent value.

We are searching for the probabily of pulling 1 or more Ash Blossom inside a single Legendary Collection, if we assume there are no short prints. To do that, we simply divide the number of the possible pulls that contains at least 1 Ash by the total number of all the possible pulls.
Inside a Legendary Collection: Kaiba there are 3 Mega Pack each containing 6 Ultra rares randomly picked amongst the total 60 available.

In order to simplify calculations we assume two hypoteses:

• It is possible to pull, inside the same Mega Pack, one or more cards in multiple copies;
• It is possible to pull multiple copies of the same card in different Mega Packs of the same Collection.

This will make our calculation to give a slightly greater probabilty than the real one, but this will not matter, as we will explain later.

Inside a single Collection we will pull 18 Ultra rares.

A quick reminder of combinatorial calculation: a k-combination of a set of n disctinct elements is the number of the possible way you can select k elements amongst the set, without the order of the selection matters. If is possible to select the same element multiple times (ex: after pulling a ticket in a raffle, you put it back in the box before pulling again) we obtain a combination with repetition.

This said, the number of the possible pulls of the 18 cards is given by the 18-combination with repetition of the set of the 60 Ultra rares :

 

The result is a number with the order of 10¹⁷.

The number pulls in which there is at least 1 Ash, will be equal to the total number of pulls we would have if the cards pulled were 17, since at least 1 out the 18 is surely an Ash. So, this number is also a combination with repetition.

 

The result is a number with the order of 10¹⁶.

We now can calculate the probability:

So, a bit less of 1 out of 4. That means that to have a reasonable possibility of pulling 1 Ash you should open a bit more than 4 boxes. IRL the Ashes are pulled even more frequently than the calculations suggested! (1 in a bit less of 4 boxes, on average). Also, the calculations were already going to give a probability higher than the real one because of the starting hypoteses.
This means that probably the Legendary Collection has one of more short prints between the Ultra rares, but Ash is not among them, and so

Ash Blossom is not short printed

That's all. To the seller and the buyers: before thinking there's a machinaction, check the facts. Reality is often more surprising than you believe.

Francesco "Francexi" Petronella