Question: Alice, Bob, and Carl arrange to meet for lunch on a certain day. They arrive independently at uniformly distributed times between 1 pm and 1:30 pm on that day.
(a) What is the probability that Carl arrives first? For the rest of this problem, assume that Carl arrives first at 1:10 pm, and condition on this fact.
(b) What is the probability that Carl will have to wait more than 10 minutes for one of the others to show up? (So consider Carl's waiting time until at least one of the others has arrived.)
(c) What is the probability that Carl will have to wait more than 10 minutes for both of the others to show up? (So consider Carl's waiting time until both of the others has arrived.)
(d) What is the probability that the person who arrives second will have to wait more than 5 minutes for the third person to show up?