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 Post subject: Re: The Mana Clinic
PostPosted: Fri Feb 07, 2020 12:56 pm 
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Thanks so far. Don't worry, I don't need to go full nerd on this, I "just" want to know my odds of getting at least one of each color in my starting hand for mulligan purposes.

Can't check out your link now but will do when I have the time.


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 Post subject: Re: The Mana Clinic
PostPosted: Fri Feb 07, 2020 6:33 pm 
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After a bit of thought your estimate of 30% isn’t likely to be too far off. It’s not accurate, but I bet it’s close. The problem is that some of the counted events will have been double counted that way. I had a pretty busy day, but I’m still thinking about an easy way to do these calculations using excel or google sheets. When I say easy, I mean easy to use - not necessarily easy to make.


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 Post subject: Re: The Mana Clinic
PostPosted: Wed Feb 12, 2020 7:17 am 
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DJ0045 wrote:
For your question, I think you want (8 choose 1)*(8 choose 0)*(24 choose 6)/(40 choose 7)

Let me englishify this for you: the first term is how many ways we can pick one mountain from 8, next is how many ways to get 0 swamps from 8 (slightly tricky concept, but the answer is just 1 by definition*), then how many ways can the rest of the deck be ordered (huge number here), divided by how many ways can the whole deck be ordered (even bigger number).

I've found the time to check out that link now.

Now that I roughly understand that example, I also understand that it doesn't cover the whole issue. This is a single fail case where we have 1 Mountain and 0 Swamps - there are more: 0/0, 2/0, 3/0 etc. I would expect each of these events to have a (slightly) different probability - right?

How do I turn "8 choose 1" into a number? Your guide says it's a binomial coefficient.

Is there a point to simplifying the problem? I mean, we could think of a bowl with 40 balls, colored black, white or red. How many different groups of 7 can you pick? Early math classes cover those things with only 2 colors. How do we get into the 3rd "dimension"?


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 Post subject: Re: The Mana Clinic
PostPosted: Wed Feb 12, 2020 11:02 am 
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https://en.wikipedia.org/wiki/Binomial_coefficient equation is right at the top.

N! Just means n times n-1 times n-2 times ... times 1. For all integers less than n.

E.g.: 4! Is 4*3*2*1 = 24, 5! = 120 etc...

So 8 choose 1 is just 8!/1!*(7!) which simplifies to 8. Which makes sense: you have 8 ways to choose 1 unique card out of 8 cards - one for each of the 8 cards.

The three dimension stuff is no different from the 2 dimension view of this math. You just account for more variables but it works the same way. The reason this is a bit more complicated, is that we are adding layers to the problem to allow for unique cards, or unique categories of cards. If you think of them as pulling colored balls from a jar, you’d get the same solutions. (E.g. land cards are red or blue balls, everything else is a black ball, the jar has 60 total balls, you remove the balls without replacing them - if I draw 7 balls how likely am I to get at least 1 red and 1 blue in the seven?).


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 Post subject: Re: The Mana Clinic
PostPosted: Wed Feb 12, 2020 11:22 am 
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Ah, I see. Excel can do x!. Should be FACT(x) in English.

Therefore I get 0,06% odds for exactly 1 Mountain and 6 non-lands from the quoted scenario.

8~1 is 8
8~0 is 1
24~6 is 134,596
40~7 is 18,643,560

Feel free to check :D

Now I gotta learn "at least x" instead of "exactly x".


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 Post subject: Re: The Mana Clinic
PostPosted: Wed Feb 12, 2020 11:27 am 
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At least x sucks, because you have to sum up the probability for each option with more than x. In your example, I think you just need 8, but it may be many more - hence the long, scratching my head period.

By the way, a quick explanation of the way these probabilities work:

The reason you can add them is that you are taking independent scenarios (note: this is not the same as the original calculation itself which incorporates many probabilities which are dependent on each other).

Think of probability as a big pie chart. The calculation you did describes just one small set of slices, but you need to add up all of the slices you want (e.g.: either all of the successes or all of the failures - the latter is usually easier), and then you can calculate the relative areas included. The relative areas will equal the probability you seek.

You can think of the probability you calculated as a Venn diagram (this is another visualization method). How likely is it that A happens given that B happens (within the Total pie chart). But you have to do it multiple times, and make sure you aren't counting the same events twice. Once you do that you sum all of those little areas you calculated and compare their size to the rest of the possible scenarios.

side note: that is exactly, in words, what the calculation you are doing does.


Last edited by DJ0045 on Wed Feb 12, 2020 1:11 pm, edited 1 time in total.

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 Post subject: Re: The Mana Clinic
PostPosted: Wed Feb 12, 2020 1:10 pm 
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Can excel do "for i=1 to n"?


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 Post subject: Re: The Mana Clinic
PostPosted: Wed Feb 12, 2020 1:12 pm 
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No - or more specifically, not without using VBA. (one of the many, many reasons why excel sucks for real mathematicians, and also why this problem is a lot harder than it should be).

BTW, this comes up way more often than you might think. I do a lot of work dealing with Financial models, and tricking Excel to do those calculations is a real pain in the butt, as soon as you want it to do anything more complicated than addition/subtraction/multiplication/or division. I usually have to do half of the work outside of excel using pen and paper and algebra/calculus/dif. eq/etc... to find discrete solutions which can then be implemented into Excel. Totally annoying, because bottom line, excel has no way to solve systems of equations or auto-optimize solutions, let alone calculate geometric series problems, like the one you are trying to solve.


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 Post subject: Re: The Mana Clinic
PostPosted: Thu Oct 22, 2020 4:22 pm 
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So I was trying to build a Yorion pile but I have big issues with the mana... I had no clue how many sources or lands I needed so I took the Karsten guidelines... took a rate and updated the info for Yorion piles... I dunno if this is actually correct but it should be at least a better guideline... DJ plz find the mistake.

My extremely complex method was to take the sources count divide it by the cards on the pile and then guess a number that got the closest to it... :teach:

Image

Image

OP updated

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Last edited by Cucho Lambreta on Thu Oct 22, 2020 10:38 pm, edited 1 time in total.

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 Post subject: Re: The Mana Clinic
PostPosted: Thu Oct 22, 2020 5:48 pm 
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I think 32-34 is about right depending on what your top end is. If your running 8 5drops (Yorion and ECD) probably 33, if your running a few 6 drops, maybe 34.

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 Post subject: Re: The Mana Clinic
PostPosted: Thu Oct 22, 2020 7:28 pm 
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I was on 38... I´m not on 38 anymore.. I think 34-36 should be correct, maybe 38 with 8+ MDFC... still what is most disturbing is the mana sources required to cast spells, that's the part where I was having more issues... specially in three color piles.

Looking at those numbers I think 3C Yorion is not such a great idea with CC or CCC on your most relevant spells.

EDIT: the % of flood is so high at 38... like you are flooding 1 out of 4 games... if you go that route you really need a thing to do with your mana... good thing that now you have to pay 3cmc to get your companion. :V :V :V

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 Post subject: Re: The Mana Clinic
PostPosted: Fri Oct 23, 2020 12:36 am 
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You can always sink your mana into castles.

Dual lands suck right now, and triomes aren't always in the 3 colors u want

The cool thing about true dual lands is, with only a couple you could reliably cast either 2WW or a 2UU spell on T4, with flip lands you often have to pick one spell or the other, making your options more limited.

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