Wednesday, December 2, 2015

1-ofs, 2-ofs, 3-ofs and 4-ofs and Hypergeometric Distribution (part 1)

Today I am starting a new miniseries here on mtgolibrary. In a Pauper Gauntlet video I critiqued a deck for having a random 1-of. That lead to an interesting discussion about the number of copies of a card to include in a deck.

Which cards can you use a single copy of? Which cards do you need all four copies of?

Before we start with the 4-ofs next week, we need to discuss the Hypergeometric Distribution. Bear with me. This will involve some light math. OK, maybe not that light. Try to bear with me anyway. To understand high level Magic theory you have to do the math. There is no other way.

Hypergeometric Distribution calculates your chances of drawing a particular number of successes from a certain population.

Imagine that you are playing Jund in Modern. You would very much like to start the game with a Tarmogoyf. What is the likelihood of drawing of having at least one Tarmogoyf in your opening hand on seven cards?

Population Size: 60 cards
Successes in Population: 4 Goyfs
Sample Size: 7 cards

Here is an online calculator for Hypergeometric Distrubitons that I like to use:

If you run the above numbers you get a 40% chance of getting at least one Goyf in your opening hand. 

Try it and see if you understand how the calculator works. 

Mulligans complicate this but the principles still hold.

We will return to Hypergeometric Distributions in the upcoming part.

I will add links below to the different parts as I publish them.

Part 2: 4-ofs Dec 9th
Part 3: 3-ofs Dec 16th
Part 4: 2-ofs Dec 23rd
Part 5: 1-ofs Dec 30th
Part 6: A sample deck 

Bonus. The even more mathematical model of doing this!

The odds of drawing a particular card in a 60-card deck are 1/60. If there are four Goyfs, the odds are 4/60. The odds of NOT drawing one of those cards in the first draw is 1 - 4/60 = 56/60.
To calculate the odds of the entire first hand, we can do it backwards:
The odds of not having any of the four cards in the first card is 56/60 The second card has odds of 55/59 (there are still 4 goyfs but only 55 non-Goyfs), and then 54/58 and so on:
  • Card 1: 56/60 chance of not being the Goyf
  • Card 2: 55/59
  • Card 3: 54/58
  • Card 4: 53/57
  • Card 5: 52/56
  • Card 6: 51/55
  • Card 7: 50/54
The odds of ALL of these happening (i.e. there is no Goyf in your opening hand) is the result of multiplying all these odds together:
(56*55*54*53*52*51*50) / (60*59*58*57*56*55*54) = 60%
If this does not happen there is at least one Goyf in your hand. That is 100%-60%= 40% likely to happen.

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