Four dogs in a row at a round table

Six dogs and six cats are placed at random in the $12$ seats around a round table. What is the probability that some four consecutive seats are all occupied by dogs?

$\tfrac{3}{11}$. A run of four or more dogs means that, going clockwise from some seat, we see a cat followed by four dogs. With only six dogs, at most one seat can start such a pattern, so these 12 events are disjoint. Each has probability $\tfrac{6}{12}\cdot\tfrac{6}{11}\cdot\tfrac{5}{10}\cdot\tfrac{4}{9}\cdot\tfrac{3}{8}=\tfrac{1}{44}$, and the total is $12\cdot\tfrac1{44}=\tfrac3{11}$.