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.
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
The fully general formula for this, if you want to get at least M mana is
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
For 4 players, this probability is 8/9.