Mana Distribution
Deck Help forum
Posted on March 31, 2021, 8:53 a.m. by epicPlain
Hi everyone
Trying to get some grip on my mana distribution decisions I found an article about hypergeometric distributions. I'm pretty convinced that I understood the theory and started to make my first calculations. Nevertheless, it seems that there is either a calculation mistake in the article or that I make a mistake. The latter is more likely to be the case - but - it would be nice to have a discussion here. Eventually, mathematicians who could explain or comment things with more details.
According to my understanding, the probability to drop x lands or more (as the first player) in the 4th turn (with a deck that has 40 cards of which 16 are lands) is achieved by summing all the probabilities (according to the hypergeometric distribution) for x, x+1,...x+n where n is the maximum number of lands that you can pick (up to turn 4); that is 10 lands after having drawn 10 cards (according to my understanding): 7 for the first hand and 3 additional in the consecutive turns being the first player.
According to my calculation, the drop 1 (so the case x=1) (in turn 4) is a probability of 99.77% but the article says 97% (which I believe is a mistake).
Can anyone confirm my statement or explain how the 97% probability comes off?
Here is a link to the article: How Not to Lose - Part One
sergiodelrio says... #3
Ah, ok. So if you're looking to find one land or more in the first 4 turns (same stats as above, except Sample size, which becomes 10) we get your 99,77%. However, that is not a useful/meaningful search setup. It accounts for all those cases where you basically sit out 1-3 turns, then draw a land.
March 31, 2021 10:41 a.m. Edited.
Thanks for the link - awesome tool! Also, thanks for the comments. Its very helpful. I still try to understand:
Not sure if I understand your second comment correctly. As far as I believe, the probability P(X ⩽ 1) is equivalent with 1-P(X > 1) because the sum of both is the sum of all probabilities: P(X ⩽ 1) + P(X > 1) = 1; in that case I still think that either I don't get it or they just made a mistake by assuming that the solution is P(X > 1) while instead it is the one that I suggest.
I'm not a native speaker, what do you mean with "sit out"?
March 31, 2021 11:12 a.m.
sergiodelrio says... #5
No problem! I'm long time out of school and pretty much forgot most formulas that apply here... that's why I use the tool :D
Sit out means you don't do anything. Here's what I mean:
To find out how you got your 99,77% value, I put the numbers I stated above in the calculator tool. Then it showed me your number.
However, what you are calculating here, correctly, is the probability finding 1 land or more after 3 turns (turn 4, on the play). But your (correct) result is impractical. Why would you like keeping 7 spells, draw a spell turn 2, discard, and then draw a land? That's what your calculation includes.
I think what you really want is:
- Population size=40
- Number of successes in population=16
- Sample size=10 (Turn 4)
- Number of successes in sample=4
= 64%
They say it themselves in the link:
"Math geek note: Actually, that is an oversimplification. Since you should mulligan no land hands, etc., the odds of randomly drawn cards being lands are slightly below that amount because of mulligans, but close enough. The math gets messy – especially if you are trying to calculate the odds of hitting your land drops, assuming you play first and have a set number of lands in your deck."
Again, I'm not a math professor, that's how I'd do it and I think that, while you calculation is technically correct, you're asking the wrong question, that's why you have a different result.
March 31, 2021 11:32 a.m.
Oh! Sure, because you basically want to have a land out every turn and that is the interesting number. I got you.
Cheers!!
sergiodelrio says... #2
Soo, not sure what you mean by
"According to my calculation, the drop 1 (so the case x=1) (in turn 4)"
But here are some results I got using this handy tool
In a 40 card deck (Population size=40) with 16 lands (Number of successes in population=16) in you opening hand (Sample size=7) the chances of drawing one land (Number of successes in sample=1), or more in our case, is 98%, rounded down.
They might have secretly discounted hands with 6 and 7 lands - that actually sums up nicely to about a whole percentage point. Those would otherwise be included in the calculation, but would not in fact produce a playable hand, and you're only looking for playable hands.
March 31, 2021 10:22 a.m.