Two shoes forming a pair

Text

From $6$ pairs of shoes, $2$ shoes are chosen at random. What is the probability that they form a pair (i.e. the left and the right of the same model)?

Solution

There are $12$ shoes in all. Choosing $2$ shoes without order gives $\binom{12}{2} = \frac{12\cdot 11}{2} = 66 \quad\text{possible cases}.$ The favorable cases are the $6$ pairs (one for each model). Hence $P = \frac{6}{66} = \frac{1}{11} \approx \boxed{0.0909}.$