MathDB

Alex, Bill, and Charlie want to play a game of DotA. They each come online at a uniformly random time between

8:00

and

8:05\text{ }\text{PM}

, and each person queues for

2

minutes. However, if any of them sees any other of them online while queuing, they merge parties and restart the queue, again waiting for

2

minutes starting from the merger time.
For example, suppose that Alex logs in at

8:00\text{ PM}

, Bill logs in at

8:01\text{ PM}

, and Charlie logs in at

8:02:30\text{ PM}

(

30

seconds past

8:02\text{ PM}

). At

8:01\text{ PM}

, Alex and Bill would merge parties and queue for

2

minutes starting at

8:01\text{ PM}

. At

8:02:30\text{ PM}

, Charlie would merge with Alex and Bill’s party, since Alex and Bill have waited together for only

1.5

minutes.
What is the probability that they will play as a party of

3

24