Six roommates are going to move into a house with four bedrooms: one double room and four single rooms. The six roommates propose that they draw names to determine the order in which they pick the bedrooms, with the assumption that the first four names drawn will choose the single rooms, so the two roommates who are not selected will need to share the double bedroom. They all put their names into a hat. Sam, Nate, Vincent, and Kyle each get to submit their names three times, while Liam and Miles only get to submit their names once. This produces a total of 14 entries. Once a name gets drawn, any duplicates of their name is removed from the hat. What is the probability that Sam is one of the first four names drawn (excluding duplicates) and will get a single room?