How do you calculate the average number of lands in your hand in a game of MTG? (This is actually a question and a half; Please tell me if this should be done otherwise)
I have a magic deck of 60 cards. Some of them (say 25) are lands. The first card I draw has a 25/60 chance of being a land. If it is, the next card has a 24/59 chance, whereas if it's not, it has a 25/59 chance. How do you find the average number of lands in your hand? (a hand is seven cards)
How do you find the standard deviation of this? How do you graph it?
 A: For the first question: Since expectation is a linear function, the expected number of lands in a 7-card hand is 7 times the expectation for any given card to be a land; for your example, this would be 7*25/60, or roughly 2.9 lands.
To go further than just this simple expectation (for instance, to find the SD and the probabilities of specific values), you'll want to look into what's known as the Hypergeometric distribution, which characterizes exactly this question; in fact, searching on 'Hypergeometric distribution magic' should give you several articles on the topic as it relates to the field of M:tG specifically.
A: Let $X$ be the random variable counting the number of lands in your starting hand (so $X$ takes on integer values between 0 and 7). Assume your deck has $d$ cards (does it always have to be 60?), $\ell$ of which are lands.
$$
\Pr(X=k) = \frac{\binom{\ell}{k}\binom{d-\ell}{7 - k}}{\binom{d}{7}}.
$$
In the numerator, I first choose exactly $k$ lands, then fill the rest of my hand with non-land cards. In the denominator, I consider all possible 7-card hands.
Once you get the probabilities for $X$ taking on all possible values (0 through 7), you can use the expected value and standard deviation formulas.
