Wednesday, March 10, 2010

Math and Magic – Probabilities

Just a short article today and two geeky probability questions.

Probabilities in Magic are very interesting but I haven't been able to use them to my advantage. If I'm playing a two color draft deck, I tend to mulligan if I don't have both of my lands (Island and Plains) but I don't explicitly use probabilities during deck construction or while playing.

The land count is always very important and probabilities might be able to indicate the number of land and land types that a deck needs. Most people develop their intuition by playing a ton of games instead of doing the (boring) calculations.

I presume probabilities would be more useful in limited games because the deck size is smaller.

If I'm splashing 3 blue spells and 5 Islands in a 40 card deck, what is the probability that I'll draw a blue spell and an Island by turn 7? (I used these numbers from Forge's Deck Analysis.)

I think the answer is 29 percent (about one-third of the time) because 0.448 x 0.639 = 0.29 (rounded to 2 decimal places).

Likewise, what is the probability of starting out with an Island and a blue spell?

The answer should be 0.125 x 0.075 = 0.01 (rounded to 2 decimal places). The answer is 1 percent of the time or 1 out of 100 games. (The super-geeky can take into account mulligans which I ignored.)

Thank you for journeying into the world-of-math. Shoutouts to the book: Godel, Escher, Bach: The Eternal Golden Braid. (It is a great book even though it twists your mind a few times. I hated the dialogue between chapters but everything else was great.)

I double checked my math but I wouldn't stake my life on it. ;-)


Forge said...

I posted the Java method generates the card drawing possibilities to Forge's Google Project so you can look at it and figure it out.

lm01 said...

Actually I don't use Forge draw calculations. They are too limited and lack details.

For instance, "draw probabilities" are for hand drawn only, not turn draws.
It seems that it calculates the % to draw at least 1 of those cards.
Probability that you draw at least an Island on 2 draws?
R: prob to draw only on 1st + prob to draw only on 2nd + prob to draw on 1st and 2nd... 5/40*35/39 + 35/40*5/39 + 5/40*4/39 = 0.237
Just like you have.

This is kind of redundant with the Random Start Hand.
The cards drawn would be great to have, but for at least 14 draws to see first 7 turns at least. This helps alot for those who want to test mana curve. With hand alone, that is not possible.

Peter said...

a. "3 blue spells and 5 Islands in a 40 card deck, what is the probability that I'll draw a blue spell and an Island by turn 7? "

b. "what is the probability of starting out with an Island and a blue spell?"

a. Is not correct, because:
1. 7th column in linked image shows probability for 7th card, not card drawn in 7th turn (which would be 14th)
2. Those are not independent events, so you cannot just multiply their probability (this is even more apparent in b. case). I guess Bayes would handle this better.

b. First column shows probabilities for SINGLE card drawn. The probability that it will be both Island and blue spell is zero, not 0.01 - this shows error in your ways :)

Huggybaby said...

Well, MWS and Nate's Deck Builder both have nice odds calculators. MWS has some nice features I haven't seen anywhere else. I'd compare a deck in all three programs to see if the results jibe.

Forge said...

I figured I was at least a little wrong with the possibility of being majorly wrong. I could have written some Java code that would loop a million times and output the correct statistics. I'll also see what Nate's Deckbuilder says.

Math (and programming) are hard to get right the first time. :)

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