First, some numbers:
Number of packs opened: 36 (1 box and half of another box)
Total number of Rares: 36 (Because each pack comes with 1 rare)
Total number of Super Rares: 5
Total number of Ultra Rares: 2
Total number of Ultimate Rares: 2
Total number of Secret Rares: 1
Total number of Ghost Rares: 0
Total number of Commons: (5+2+2+1) * 7 + (36 - (5+2+2+1)) * 8 = 278
Total number of Cards: 36 * 9 = 324
We can only calculate the number of commons after we find out how many of higher rarities there are. Since each pack is at worst a rare, those packs will have 8 commons and 1 rare. Packs with a card higher than a rare will also come with a rare; hence the 7 commons.
Most frequent Common: Token Stampede (10)
Least frequent Common: Xing Zhen Hu Replica (2)
Most frequent Rare: Guldfaxe of the Nordic Beasts & Legendary Six Samurai - Kageki (3)
Least frequent Rare: All rares appeared at least once; so no least frequent.
Doing some quick math, you can confirm that this is indeed a proper pmf, as all the p(x) add up to 1.
I'd like to note that because of our small sample space, I didn't pull a ghost rare (As an aside, my friend who opened the other half of the box did pull the Ghost Odin; if I were to consider his pulls, the data set would be influenced greatly. But I will not for the sake of simplicity).
Using this sample space, we can infer some statistics:
- 72% of all packs you open will only contain a rare (Not a promising number, I know, but that's how they make money)
- No rares are short-print (Seeing as I got at least one of each rare)
- Certain cards are short-print (A hypothesis, albeit one I cannot perform hypothesis testing on)
- The distribution of rarity is correct (In the sense that you'll get more Supers than Ultras, etc). We can infer that if you opened enough packs (Enough is subjective, but we're talking 100+ for consistency purposes)
- Ultra, Ultimate, and Ghost Rare share the same probability distribution (Once again, due to the nature of this small sample space, it is inconclusive, but it is again an inference).
For our pmf, we had to introduce the clause "only" for the rare rarity. Since you always get a rare, then there would be a 100% chance of getting a rare, which would not follow the rules of a pmf.
If we assume that no rares are short-print, then there is a 5% (1/20, because there are 20 rares in this set) of getting a particular rare you want. Since you always get a rare every pack, there is no conditional probability. If we were, for some reason, interested in the probability of getting a particular rare in a pack that only contained a rare, that probability would instead be 3.61%.
Once again, if we assume that no super rares are short-print, we find that our probability of getting a particular super rare is ever so slightly less than 1% (0.992%). Since this is an Intersection (Getting a Super Rare AND getting a certain Super Rare), we find this by multiplying the probability of obtaining a super rare (0.1389) with the probability of getting a particular super rare (1/14 = 0.0714). This is assuming there is no short-print factor; if so, then pulling a Kizan as I did would be even more rare (And the fact that my friend pulled one as well is a ridiculous probability; then again, we can assume independence).
Going to the Ultra/Ultimate/Ghost trio is probably the most interesting of the data set (I'll explain why later). Let's first calculate the probability of getting any one of the aforementioned, which is 11.12% (Since this is an Union (Ultra OR Ultimate) condition, we add all probabilities). We expect that this probability is less than that of obtaining a super; which we find to be true.
Now here is the interesting part. Suppose we wanted to calculate the probability of pulling the Ultimate Rare "Legendary Six Samurai - Enishi". If we assume that there is an equal chance of obtaining Ultra or Ultimate, then our probability is therefore 0.556% (Another Intersection; multiply the probability of an Ultra/Ultimate with the probability of getting Enishi).
Although this pmf doesn't have any probability for Ghost Rare, we can infer that it would share the total probability of getting the trio (So the Ghost Rare probability becomes 0.1112 / 3 = 0.0371). This is a special case, since Ghost Rare is exclusive to one card per set. Let's calculate that probability:
P(Ultra/Ultimate/Ghost ∪ Odin ∪Ghost) = (0.1112) * (1/10) * (1/3)= 0.00317 or 0.371%
This probability would also apply to the other 2 rarities of Odin.
So there's a 0.371% of getting an Odin in a pack. Why is this interesting? Once we examine the Secret Rare set, it will become clear.
Finally, we do the same thing for Secret Rare (Probability of getting a Secret Rare and getting a particular one), and we get 0.003475 or 0.3475%
What does this data tell us? It infers that there is a lower probability of getting ANY particular Secret Rare card than it is to get a Ghost Rare Odin. I repeat one last time that this is only an inference, and we cannot conclude this as a fact.
Eventually (After exams most likely), I might continue with a post relating the probability of making money by opening packs. This all has to do with Expected Value (More commonly known as the mean). We'll see how much I'll hate Statistics after the semester.
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