Okay, so many many many many...what is it, weeks? Maybe a month? Anyway, about the time of my Grade Ratio article or, shortly afterwards, I was told by a buddy over at Vmundi that their friend totally scrapped my numbers because I never showed my math.

May I first start out with screw you Vmundi buddy's friend

Okay, so this article is probably going to boor the absolute hell out of a lot of you, and I really won't blame you. But, for those of you in high school Statistics class who want a refresher with polynomial or hypergeometric distribution, or simply just want your own way to create very solid claims backed by solid math evidence, you may want to listen up.

To start this off, I used a calculator. *Throws up shield to defend self*

Okay, are all the critics gone? Anyway, I actually used 2 calculators. The first being a hypergeometric calculator from what I can get of what it's called, and a more familiar looking scientific calculator. You can probably call me out for better, more accurate calculators later but for now, here's what it is.

The first calculator simplifies everything for me. Let's say of 4 grade 1s I run, I want to draw in my opening hand at least 1, so I plug in the deck amount as my population size (49), the 4 grade 1s as my successes (4), drawing my opening hand as the sample size (5), and looking for atleast 1 success (1). Basically, plug these all in and you can get?

If you followed me, X<1 should be 0.640709660367385, or 1221759/1906884 if you want a fraction. You'll notice that I am now focusing on the chance for failure. Little note for all of you soon to be scholars, when you are looking for the chances for 'atleast 1', you look for the chances of absolute failure, and subtract those from 1. But, in order to do that, we must finish this.

Now, because I consider mulligan 3 the 'standard' (if you are mulliganing and still worrying about a certain Grade, you are most definitely mulliganing at least 3.) We return 3 cards to the deck and start over. With a population of now 47, still 4 successes in population because we were supposed to fail drawing any to begin with, sample size 4 for the 3 redraws and the first turn draw, and still just 1 success, you should find X<1. Plug this in right and you should have 0.691895831581308, or 24682/35673.

NOW we can get these right. The reason behind the fractions is because despite how accurate the hypergeometric calculator is, its still rounding off. With fractions with whole numbers in them, it's just that much easier to get a more accurate number. So, I think we all know how to multiply fractions here.

Basically, the final answer should be 0.44330434326207, or 3108169/7011366. So, we can say ~44% of the time, we get one of those 4 Grade 1s in our opening hand right? Wrong.

Recall a little something something I mentioned earlier. I said we are multiplying the absolute failures together. Meaning, ~44% of the time is the actual failure to grab a card. The real percentage we are looking for here is 3903197/7011366, or 0.55669565673793, meaning ~56% of the time you will draw them.

With basic reasoning, you can see how I got the later Grade 2s and Grade 3s using the assumptions I mentioned. You could also see why I don't want to finish this because even though I already finished the finer calculations, and despite how simple this is, it's quite time consuming (at least for me and my attention span) and thus answers why I never finished.

Alright, at this point, I don't know what your brain is going through, so here's a puppy and kitty I found on Google. (scroll down, I didn't leave this any closer because it might have distracted you, also this image is just like that.

Okay, back?

Now, here's a test for YOU. I already have the numbers for CEO Amaterasu's Trigger success ratios on a Word Document (lolWord) that has the basic gist of everything. The test question is this:

Assuming you do not Soulcharge a trigger, and there are now 12 triggers left in a...oh sorry, the puppy and kitty are now so close to what I'm typing I got lost for a moment.

Tiny bit farther away now, assuming you do not Soulcharge a trigger, and you are left with a 37 card deck with 12 triggers, what is the probability of pulling a trigger. This includes between stacking a trigger your topdeck might reveal and leave it on the top, and moving a nontrigger from your topdeck to the bottom, calculating both.

Okay, that actually sounds really hard to count so I'll simplify this into 3 questions.

1)With a 38 card, 12 trigger deck, what is the likeliness that you will Twin Drive into 0, 1, and 2 triggers?

2)With a 38 card, 12 trigger deck, and CEO does not soulcharge a trigger, what is the likeliness of her Twin Drive checking either 0, 1, or 2 triggers after calculating her divination?

3)With a 38 card, 12 trigger deck, what is the likeliness of CEO Amaterasu triggering into 0, 1, or 2 triggers after you factor the divination, AND the what happens if she Soulcharges a trigger.

No real prizes for answering these correctly except for probably expanded knowledge, bragging rights, me mentioning you a few more times later down the road and hopefully a better mathematical edge over everyone else you know when it comes to probability, but hey, if you can answer something like the three questions correctly, you are probably more than able to mathematically back up any claims you so choose when it comes to probability in card games.

David i'm your biggest fan marry me plx

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