The team does not randomly pick people, the computer picks locations and times. This makes it fair for everyone traveling alone, with kids, adults only, doesn't matter it is not a person looking for you. With the prizes like the lanyards, ears, fastpasses they are given out in bulk so the team will get a slip that says something like At 9:30 issue 50 Dream Fastpasses to the guests loading Dumbo. At exactlly 9:30 they will be at the Dumbo turnstile issuing the passes to the next 50 people that come through. If it is a big prize it will be VERY specific. It will say at 2:20pm issue Castle Stay (or whatever) to the guest in row 4 left seat on the Rockin Roller Coaster. The train to come into the statino AT or just after 2:20 that person in row 4 left seat will be the winner. If there is no one in that seat, they will put the prize back into the computer to be distributed at a new time and location later in the day (or just later if it is not a day specific prize).
So your travelin party really has nothing to do with how the prizes are won. Single, family reunion, Adults, kids... everyone is fair game....