Hypergeometry of counterfeits

A collection of 15 gold coins contains 4 counterfeits. If 2 of them are randomly selected to be sold at the auction, find the probability that

  1. neither of them is a counterfeit
  2. only one of them is a counterfeit
  3. both coins are counterfeits.

This is a hypergeometric probability distribution – picking without replacement. If X is the number of counterfeit coins (hypergeometric random variable),

P(X = 0) = \frac{_{4}C_0 \textrm{ }*\textrm{ } _{11}C_2\textrm{ }}{_{15}C_2}

choose(4,0)*choose(11,2) / choose(15,2)
0.52

P(X = 1) = \frac{_{4}C_1 \textrm{ }*\textrm{ } _{11}C_1\textrm{ }}{_{15}C_2}

choose(4,1)*choose(11,1) / choose(15,2)
0.42

P(X = 2) = \frac{_{4}C_2 \textrm{ }*\textrm{ } _{11}C_0\textrm{ }}{_{15}C_2}

choose(4,2)*choose(11,0) / choose(15,2)
0.06

Or simply,

dhyper(2, 4, 11, 2, log = FALSE)

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