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Math Help - Card game probability question

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    Card game probability question

    I've not done probability in a while and looking at online examples I don't know if I'm doing everything right so maybe you can can help me out with the formula and explanation.

    I'm a big fan of the game Magic the Gathering and when playing the game I like a healthy number of "land" cards in my opening hand. There are 60 cards in my deck and 26 land cards I've put in. What is the probability I will have 3 land cards in my opening hand (7 cards)? What about 2 land cards? What about atleast 2 land cards? Thanks in advance.
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    Quote Originally Posted by XIII13Thirteen View Post
    I've not done probability in a while and looking at online examples I don't know if I'm doing everything right so maybe you can can help me out with the formula and explanation.

    I'm a big fan of the game Magic the Gathering and when playing the game I like a healthy number of "land" cards in my opening hand. There are 60 cards in my deck and 26 land cards I've put in. What is the probability I will have 3 land cards in my opening hand (7 cards)? What about 2 land cards? What about atleast 2 land cards? Thanks in advance.
    Since I don't know anything about the game, would you clarify the problem statement, please?

    Is the total number of card in the deck 60 (26 land plus 34 non-land), or 86 (60 non-land plus 26 land)?
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    Quote Originally Posted by awkward View Post
    Since I don't know anything about the game, would you clarify the problem statement, please?

    Is the total number of card in the deck 60 (26 land plus 34 non-land), or 86 (60 non-land plus 26 land)?
    Sorry, I'll clarify. Also I apologize if I placed this in the wrong forum. I saw the basic probability forum questions above just today. Anyway, onto the clarification.

    Total cards in the deck: 60 (26 land and 34 non-land)
    Opening hand: 7 cards
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    Quote Originally Posted by XIII13Thirteen View Post
    Sorry, I'll clarify. Also I apologize if I placed this in the wrong forum. I saw the basic probability forum questions above just today. Anyway, onto the clarification.

    Total cards in the deck: 60 (26 land and 34 non-land)
    Opening hand: 7 cards
    Let's say the number of land cards in your hand is x. Then we are dealing with a hypergeometric distribution, and

    p(x) = \frac{\binom{26}{x}  \binom{34}{7-x}} { \binom{60}{7}}

    For x = 0 to 7, this yields p(x) =
    0.0139, 0.0905, 0.2342, 0.3122, 0.2316, 0.0956, 0.0203, 0.0017.

    To find the probability that you have at least 2 land cards, you could either add up p(x) for x = 2 to 7, or you could save computation by using

    P(X \ge 2) = 1 - P(X = 0) - P(X = 1).
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