Three couples go to the movie theater. They want to seat people together for maximum enjoyment, but instead they randomly file into a row with six seats. What is the probability that they sit in a socially optimal configuration, in which each person is sitting next to his or her partner?
There are $\binom{6}{2} = 15$ ways to pick people to fill the first two seats, and 3 ways for those two people to be a couple, for a probability $3/15 = 1/5$ that the first two seats are a couple. If a couple is seated successfully, there are $\binom{4}{2} = 6$ ways to seat people in the next two seats, and 2 ways for those two people to be a couple (you could pick either of the two remaining couples), for a probability $2/6 = 1/3$ that those two seats are a couple. If the first two couples are seated successfully, the last two seats will be guaranteed to go to the last couple. So, the probability that everything goes right is $1/5 \cdot 1/3 = \boxed{\frac{1}{15}}$.