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I am trying to figure out what are the chance if getting 2 or more mana after parleying Selvala, Explorer Returned, assuming that using commander/EDH one third of the deck is land and that at least four people are playing. Mainly curious after my friends mentioned way to go infinite if I could reliably generate mana.

  • I'm not sure how "infinite" it would be, since everyone will be drawing cards. At some point people will run out of cards and lose. – Becuzz Apr 21 '16 at 18:20
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    The answer to this question also depends on how many people you are playing with. – Becuzz Apr 21 '16 at 18:21
  • Lets assume its a four player game, and I know it won't be infinite per say I just wanted to know the probability – Peter Kwiatek Apr 21 '16 at 18:24
  • I think that assuming that only 1/4 of a deck is land in EDH is unrealistic. Going below 1/3 is already risky, and 2/5 is common. – murgatroid99 Apr 21 '16 at 18:24
  • I said assuming a one land to 3 nonland ratio, not one to four – Peter Kwiatek Apr 21 '16 at 18:30
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In the situation you describe, each player reveals their card independently, and you get mana equal to the number of non-land cards revealed. This is called a Binomial Distribution, with N = 4 and P = 2/3 in your particular case. So, the probability of getting exactly k mana is (4 choose k) * (2/3) ^ k * (1/3) ^ (n - k). You are specifically looking for the probability of getting at least a certain amount of mana, so you are looking for the Cumulative Binomial Distribution. To get the probability of getting at least a certain amount of mana, you simply add up the probabilities for each amount that is at least that number. So, to calculate that for your particular situation, you would calculate

Sum from k = 2 to 4 of (4 choose k) * (2 / 3) ^ k * (1 / 3) ^ (4 - k)

The fully general formula for this, if you want to get at least M mana is

Sum from k = M to N of (N choose k) * (P) ^ k * (1 - P) ^ (N - k)

In the particular case of k = 2, we can instead simply consider the cases where you don't get at least 2 mana, and then subtract that total probability from 1. For each player, the probability of only that player flipping a non-land is 2/3 * (1/3) ^ (N - 1). The probability of every player flipping a land is (1/3) ^ N. So, the total probability of getting at least two mana is

1 - (N * 2/3 * (1/3) ^ (N - 1) + (1/3) ^ N)

For 4 players, this probability is 8/9.

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    Thanks this is what I was looking for also, how would you use this to figure out the probability of generating x mana – Peter Kwiatek Apr 21 '16 at 19:01
  • I added a fully general formula – murgatroid99 Apr 21 '16 at 19:46
  • What does ^ mean? – Zinma Apr 22 '16 at 0:28
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    That's exponentiation. We don't have MathJax on this site, so I don't really have a better way to represent it. – murgatroid99 Apr 22 '16 at 0:57
  • You could use a Latex generator to create a static image: arachnoid.com/latex/… for middle equation, arachnoid.com/latex/… for the last one. – doppelgreener Apr 24 '16 at 6:55

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