Question: Tyrion, Cersei, and n other guests arrive at a party at i.i.d. times drawn from a continuous distribution with support [0, 1], and stay until the end (time 0 is the party's start time and time 1 is the end time). The party will be boring at times when neither Tyrion nor Cersei is there, fun when exactly one of them is there, and awkward when both Tyrion and Cersei are there.
(a) On average, how many of the n other guests will arrive at times when the party is fun?
(b) Jaime and Robert are two of the other guests. By computing both sides in the definition of independence, determine whether the event "Jaime arrives at a fun time" is independent of the event "Robert arrives at a fun time".
(c) Give a clear intuitive explanation of whether the two events from (b) are independent, and whether they are conditionally independent given the arrival times of everyone else, i.e., everyone except Jaime and Robert.