Poker in Magic, part 1/4

This article series will be about constituent of poker in Magic the Gathering. Poker is not a game where luck plays first fiddle, but rather pure skill. Otherwise we will not see the same faces during Poker Word Series and our chances of winning against pro player should be equal 50/50, but I guarantee you they are not! I would like to focus on three basic matters: probability, bluff, tells(reading the signs what our opponent might have on his hand). On the grounds of vast problem I decided to divide it into few, separate articles.

Probability
Consider that in a 60-card deck there are roughly 2 trillion opening hands you can draw, and that’s before turn one! Even accounting for the degeneracy built into most high-level decks, in the large number of basic lands and the rule of four for many of the most powerful cards, we can still expect that there are more combinations, simply in the opening hand, than even the most devout play testers are capable of seeing, no matter how many times they test a particular build.
Let’s do the simplest mathematical calculation. Assume, deck consists 20 lands and 40 spells. We draw seven cards. The odds of getting X lands in an opening hand of 7 cards would be given by the hypergeometric distribution function presented below:


Okay assume, we have 1 land and 6 other spells with converted mana cost 2-3, which are good in this matchup. If we draw a land in early turn we will win. Should we keep such a hand or take a mulligan?
Let’s do some math. What do we know? In library left 53 cards. Out of these 53 cards 19 cards are lands. In order to count probability of drawing a land we must use this formula: P=lands left in library / total number of cards left in library. Our chances to draw a land are equal P= 19/53 = 35.84%,

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