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Red cards in a hand

ProbabilityMedium

A standard deck has $52$ cards, of which $26$ are red. You deal $3$ cards uniformly at random without replacement. What is the probability that exactly $2$ of them are red? Give a fraction in lowest terms.

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Hint

Count favorable hands with binomial coefficients and divide by all possible hands.

Solution

The number of all hands is $(3 52) = 22100$ .
Choose the $2$ red cards in $(2 26) = 325$ ways.
Choose the remaining non-red card in $26$ ways, so there are $325 \times 26 = 8450$ favorable hands.
The probability is $\frac{8450}{22100}$ , which reduces to $\frac{13}{34}$ .

Answer: $\frac{13}{34}$

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