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?

• When you say "in your hand", do you mean "in your starting hand"? My understanding is that you put them into play, so that eventually you will not be drawing any. – The Chaz 2.0 May 17 '12 at 17:50
• Also, check out the "Board and Card Games" stack exchange site, which is mostly MtG questions. – The Chaz 2.0 May 17 '12 at 18:09
• @The Chaz: Yes. Also, draw3cards.com is so much better for that. And I doubt they answer math questions. – Glycan May 18 '12 at 20:06

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.

• This isn't exactly what I was asking, but it's close enough, and constructive. – Glycan May 18 '12 at 20:28

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.

• To answer the parenthetical: the minimum for constructed play is 60 (though the question also makes sense for 40-card decks, which are the minimum allowed in most limited play); most people choose to play as closely to that minimum as possible to minimize variance. (Though I've seen far too little attention paid to just how much abstract expectation one is giving up by going one or two cards over the minima) – Steven Stadnicki May 17 '12 at 18:02
• This is really confusing. It's probably righter than the other answer, but the other answer was more constructive. – Glycan May 18 '12 at 20:28
• @GLycan What part can I help clarify? – Austin Mohr May 18 '12 at 20:44
• @AustinMohr: I don't understand the notation, at all, although that's a lacking on my part, not yours. – Glycan May 19 '12 at 1:46
• @GLycan $\Pr(X=k)$ reads "the probability of having exactly $k$ lands". The right-hand side of the equation computes this probability using binomial coefficients. These numbers are sometimes written, for example, $\ell$C$k$ (especially on calculators), but $\binom{\ell}{k}$ is more standard. – Austin Mohr May 19 '12 at 1:55